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Distributed parameter system

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In <a href="Control_theory" title="Control theory">control theory</a>, a distributed-parameter system (as opposed to a <a href="Lumped-element_model" title="Lumped-element model">lumped-parameter system</a>) is a <a href="System" title="System">system</a> whose <a href="State_space_(controls)" class="mw-redirect" title="State space (controls)">state space</a> is infinite-<a href="Dimension_(vector_space)" title="Dimension (vector space)">dimensional</a>. Such systems are therefore also known as infinite-dimensional systems. Typical examples are systems described by <a href="Partial_differential_equation" title="Partial differential equation">partial differential equations</a> or by <a href="Delay_differential_equation" title="Delay differential equation">delay differential equations</a>.

<div class="mw-heading mw-heading2"><h2 id="Linear_time-invariant_distributed-parameter_systems">Linear time-invariant distributed-parameter systems</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Abstract_evolution_equations">Abstract evolution equations</h3></div>

<div class="mw-heading mw-heading4"><h4 id="Discrete-time">Discrete-time</h4></div>

With U, X and Y <a href="Hilbert_space" title="Hilbert space">Hilbert spaces</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\,}">

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<annotation encoding="application/x-tex">{\displaystyle A\,}</annotation>

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</math><img src="./c6aaf5ce10d6add44b973e28fb3d95f37abf3721.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.13ex; height:2.176ex;" alt="{\displaystyle A\,}" loading="lazy"> ∈ L(X), <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B\,}">

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<annotation encoding="application/x-tex">{\displaystyle B\,}</annotation>

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</math><img src="./d8a72cbbfdbb8b9d0dad053538c330994b308bae.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.151ex; height:2.176ex;" alt="{\displaystyle B\,}" loading="lazy"> ∈ L(U, X), <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C\,}">

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<annotation encoding="application/x-tex">{\displaystyle C\,}</annotation>

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</math><img src="./785a192e3331793e37b1be0c5315d196da1a7049.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.153ex; height:2.176ex;" alt="{\displaystyle C\,}" loading="lazy"> ∈ L(X, Y) and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D\,}">

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<annotation encoding="application/x-tex">{\displaystyle D\,}</annotation>

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</math><img src="./1393ae1816376a473758d4b30cd5f36b8823eecc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.311ex; height:2.176ex;" alt="{\displaystyle D\,}" loading="lazy"> ∈ L(U, Y) the following <a href="Difference_equation" class="mw-redirect" title="Difference equation">difference equations</a> determine a discrete-time <a href="Linear_time-invariant_system" title="Linear time-invariant system">linear time-invariant system</a>:

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<annotation encoding="application/x-tex">{\displaystyle x(k+1)=Ax(k)+Bu(k)\,}</annotation>

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</math><img src="./5e691e6bd10c4f37c375c0e6e1a32dd275c90cb7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.887ex; height:2.843ex;" alt="{\displaystyle x(k+1)=Ax(k)+Bu(k)\,}" loading="lazy"></dd>

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<annotation encoding="application/x-tex">{\displaystyle y(k)=Cx(k)+Du(k)\,}</annotation>

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</math><img src="./3e4c1db82037e58ecf7b0f6cde8235c099e69efa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.893ex; height:2.843ex;" alt="{\displaystyle y(k)=Cx(k)+Du(k)\,}" loading="lazy"></dd></dl>

with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\,}">

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<annotation encoding="application/x-tex">{\displaystyle x\,}</annotation>

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</math><img src="./ab34739435d9d9d99cddf4041740b107343b1398.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.717ex; height:1.676ex;" alt="{\displaystyle x\,}" loading="lazy"> (the state) a sequence with values in X, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u\,}">

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<annotation encoding="application/x-tex">{\displaystyle u\,}</annotation>

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</math><img src="./880f91e25cd451d89d1f6d0d06852b56a7b74a32.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.717ex; height:1.676ex;" alt="{\displaystyle u\,}" loading="lazy"> (the input or control) a sequence with values in U and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y\,}">

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<annotation encoding="application/x-tex">{\displaystyle y\,}</annotation>

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</math><img src="./1c8c233e7cc39fac816991250d86e09b515d02e0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.543ex; height:2.009ex;" alt="{\displaystyle y\,}" loading="lazy"> (the output) a sequence with values in Y.

<div class="mw-heading mw-heading4"><h4 id="Continuous-time">Continuous-time</h4></div>

The continuous-time case is similar to the discrete-time case but now one considers differential equations instead of difference equations:

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<annotation encoding="application/x-tex">{\displaystyle {\dot {x}}(t)=Ax(t)+Bu(t)\,}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle y(t)=Cx(t)+Du(t)\,}</annotation>

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</math><img src="./b74d34d879cb6e7448c281da11706956b6112957.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.778ex; height:2.843ex;" alt="{\displaystyle y(t)=Cx(t)+Du(t)\,}" loading="lazy">.</dd></dl>

An added complication now however is that to include interesting physical examples such as partial differential equations and delay differential equations into this abstract framework, one is forced to consider <a href="Unbounded_operator" title="Unbounded operator">unbounded operators</a>. Usually A is assumed to generate a <a href="C0_semigroup" class="mw-redirect" title="C0 semigroup">strongly continuous semigroup</a> on the state space X. Assuming B, C and D to be bounded operators then already allows for the inclusion of many interesting physical examples,<a href="#cite_note-1">[1]</a> but the inclusion of many other interesting physical examples forces unboundedness of B and C as well.

<div class="mw-heading mw-heading3"><h3 id="Example:_a_partial_differential_equation">Example: a partial differential equation</h3></div>

The partial differential equation with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t>0}">

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<annotation encoding="application/x-tex">{\displaystyle t>0}</annotation>

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</math><img src="./29a2960e88369263fe3cfe00ccbfeb83daee212a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.101ex; height:2.176ex;" alt="{\displaystyle t>0}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \xi \in [0,1]}">

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<annotation encoding="application/x-tex">{\displaystyle \xi \in [0,1]}</annotation>

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</math><img src="./192670ccd7d161628924111b3854842d8cca3c2f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.523ex; height:2.843ex;" alt="{\displaystyle \xi \in [0,1]}" loading="lazy"> given by

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<annotation encoding="application/x-tex">{\displaystyle {\frac {\partial }{\partial t}}w(t,\xi )=-{\frac {\partial }{\partial \xi }}w(t,\xi )+u(t),}</annotation>

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</math><img src="./a743168aab3df5ad1fc380468bdc0023c33f3e50.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:31.305ex; height:6.009ex;" alt="{\displaystyle {\frac {\partial }{\partial t}}w(t,\xi )=-{\frac {\partial }{\partial \xi }}w(t,\xi )+u(t),}" loading="lazy"></dd>

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<annotation encoding="application/x-tex">{\displaystyle w(0,\xi )=w_{0}(\xi ),}</annotation>

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</math><img src="./77eeaa6847e65b1400a7799c560c642e56caea87.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.003ex; height:2.843ex;" alt="{\displaystyle w(0,\xi )=w_{0}(\xi ),}" loading="lazy"></dd>

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<annotation encoding="application/x-tex">{\displaystyle w(t,0)=0,}</annotation>

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</math><img src="./bbc6d47b6aa6874f942ac453a26959c7e573e198.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.417ex; height:2.843ex;" alt="{\displaystyle w(t,0)=0,}" loading="lazy"></dd>

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<annotation encoding="application/x-tex">{\displaystyle y(t)=\int _{0}^{1}w(t,\xi )\,d\xi ,}</annotation>

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</math><img src="./8cfee908410e944ac974f07317bda687d0ccf8ba.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:20.465ex; height:6.176ex;" alt="{\displaystyle y(t)=\int _{0}^{1}w(t,\xi )\,d\xi ,}" loading="lazy"></dd></dl>

fits into the abstract evolution equation framework described above as follows. The input space U and the output space Y are both chosen to be the set of complex numbers. The state space X is chosen to be L2(0, 1). The operator A is defined as

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<annotation encoding="application/x-tex">{\displaystyle Ax=-x'}</annotation>

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</math><img src="./91ac648dc38f15980df200e8b0eb49654147fdd4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.994ex; height:2.676ex;" alt="{\displaystyle Ax=-x'}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D(A)=\left\{x\in X:x{\text{ absolutely continuous }},\,x'\in L^{2}(0,1),\,x(0)=0\right\}.}">

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<mtext> absolutely continuous </mtext>

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<annotation encoding="application/x-tex">{\displaystyle D(A)=\left\{x\in X:x{\text{ absolutely continuous }},\,x'\in L^{2}(0,1),\,x(0)=0\right\}.}</annotation>

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</math><img src="./a2edfa0189c5db5d9312a6855468facc77227800.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:68.823ex; height:3.343ex;" alt="{\displaystyle D(A)=\left\{x\in X:x{\text{ absolutely continuous }},\,x'\in L^{2}(0,1),\,x(0)=0\right\}.}" loading="lazy"></dd></dl>

It can be shown<a href="#cite_note-2">[2]</a> that A generates a strongly continuous <a href="Semigroup" title="Semigroup">semigroup</a> on X. The bounded operators B, C and D are defined as

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<annotation encoding="application/x-tex">{\displaystyle Bu=u,~~~Cx=\int _{0}^{1}x(\xi )\,d\xi ,~~~D=0.}</annotation>

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</math><img src="./d26e4f26e1a62587977a1d89512c3f660bb4cb97.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:36.808ex; height:6.176ex;" alt="{\displaystyle Bu=u,~~~Cx=\int _{0}^{1}x(\xi )\,d\xi ,~~~D=0.}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading3"><h3 id="Example:_a_delay_differential_equation">Example: a delay differential equation</h3></div>

The delay differential equation

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<annotation encoding="application/x-tex">{\displaystyle {\dot {w}}(t)=w(t)+w(t-\tau )+u(t),}</annotation>

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</math><img src="./2bd19c517c3a7bd8ab53427c45d796fa6d31da13.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.386ex; height:2.843ex;" alt="{\displaystyle {\dot {w}}(t)=w(t)+w(t-\tau )+u(t),}" loading="lazy"></dd>

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<annotation encoding="application/x-tex">{\displaystyle y(t)=w(t),}</annotation>

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</math><img src="./dd2f2d7ec15522fd288c080aa37a3f49cf01a7fe.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.863ex; height:2.843ex;" alt="{\displaystyle y(t)=w(t),}" loading="lazy"></dd></dl>

fits into the abstract evolution equation framework described above as follows. The input space U and the output space Y are both chosen to be the set of complex numbers. The state space X is chosen to be the product of the complex numbers with L2(−τ, 0). The operator A is defined as

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<annotation encoding="application/x-tex">{\displaystyle A{\begin{pmatrix}r\\f\end{pmatrix}}={\begin{pmatrix}r+f(-\tau )\\f'\end{pmatrix}}}</annotation>

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</math><img src="./cdc140d16a764a59c853fcd00103a240c49acd0e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:24.452ex; height:6.176ex;" alt="{\displaystyle A{\begin{pmatrix}r\\f\end{pmatrix}}={\begin{pmatrix}r+f(-\tau )\\f'\end{pmatrix}}}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D(A)=\left\{{\begin{pmatrix}r\\f\end{pmatrix}}\in X:f{\text{ absolutely continuous }},\,f'\in L^{2}([-\tau ,0]),\,r=f(0)\right\}.}">

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<mtext> absolutely continuous </mtext>

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<annotation encoding="application/x-tex">{\displaystyle D(A)=\left\{{\begin{pmatrix}r\\f\end{pmatrix}}\in X:f{\text{ absolutely continuous }},\,f'\in L^{2}([-\tau ,0]),\,r=f(0)\right\}.}</annotation>

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</math><img src="./cc464a74f648d4e1efd9c09e2c179deac57630c9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:76.636ex; height:6.176ex;" alt="{\displaystyle D(A)=\left\{{\begin{pmatrix}r\\f\end{pmatrix}}\in X:f{\text{ absolutely continuous }},\,f'\in L^{2}([-\tau ,0]),\,r=f(0)\right\}.}" loading="lazy"></dd></dl>

It can be shown<a href="#cite_note-3">[3]</a> that A generates a strongly continuous semigroup on X. The bounded operators B, C and D are defined as

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<annotation encoding="application/x-tex">{\displaystyle Bu={\begin{pmatrix}u\\0\end{pmatrix}},~~~C{\begin{pmatrix}r\\f\end{pmatrix}}=r,~~~D=0.}</annotation>

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</math><img src="./989092731fe86af39f5274d6a2a2e3d0d5e8aed6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:35.443ex; height:6.176ex;" alt="{\displaystyle Bu={\begin{pmatrix}u\\0\end{pmatrix}},~~~C{\begin{pmatrix}r\\f\end{pmatrix}}=r,~~~D=0.}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading3"><h3 id="Transfer_functions">Transfer functions</h3></div>

As in the finite-dimensional case the <a href="State_space_(controls)" class="mw-redirect" title="State space (controls)">transfer function</a> is defined through the <a href="Laplace_transform" title="Laplace transform">Laplace transform</a> (continuous-time) or <a href="Z-transform" title="Z-transform">Z-transform</a> (discrete-time). Whereas in the finite-dimensional case the transfer function is a proper rational function, the infinite-dimensionality of the state space leads to irrational functions (which are however still <a href="Holomorphic_function" title="Holomorphic function">holomorphic</a>).

<div class="mw-heading mw-heading4"><h4 id="Discrete-time_2">Discrete-time</h4></div>

In discrete-time the transfer function is given in terms of the state-space parameters by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D+\sum _{k=0}^{\infty }CA^{k}Bz^{k}}">

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<annotation encoding="application/x-tex">{\displaystyle D+\sum _{k=0}^{\infty }CA^{k}Bz^{k}}</annotation>

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</math><img src="./d0c5491ab35be3bec4b50451c018d540181bada8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:17.048ex; height:7.009ex;" alt="{\displaystyle D+\sum _{k=0}^{\infty }CA^{k}Bz^{k}}" loading="lazy"> and it is holomorphic in a disc centered at the origin.<a href="#cite_note-4">[4]</a> In case 1/z belongs to the resolvent set of A (which is the case on a possibly smaller disc centered at the origin) the transfer function equals <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D+Cz(I-zA)^{-1}B}">

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<annotation encoding="application/x-tex">{\displaystyle D+Cz(I-zA)^{-1}B}</annotation>

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</math><img src="./2045eba04013e9f15c9c2893b9dbdea3f80cd1f1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.369ex; height:3.176ex;" alt="{\displaystyle D+Cz(I-zA)^{-1}B}" loading="lazy">. An interesting fact is that any function that is holomorphic in zero is the transfer function of some discrete-time system.

<div class="mw-heading mw-heading4"><h4 id="Continuous-time_2">Continuous-time</h4></div>

If A generates a strongly continuous semigroup and B, C and D are bounded operators, then<a href="#cite_note-5">[5]</a> the transfer function is given in terms of the state space parameters by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D+C(sI-A)^{-1}B}">

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<annotation encoding="application/x-tex">{\displaystyle D+C(sI-A)^{-1}B}</annotation>

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</math><img src="./6018f15ee7c02daef48ca6029e6f35b465eb9e9d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.283ex; height:3.176ex;" alt="{\displaystyle D+C(sI-A)^{-1}B}" loading="lazy"> for s with real part larger than the exponential growth bound of the semigroup generated by A. In more general situations this formula as it stands may not even make sense, but an appropriate generalization of this formula still holds.<a href="#cite_note-6">[6]</a>

To obtain an easy expression for the transfer function it is often better to take the Laplace transform in the given differential equation than to use the state space formulas as illustrated below on the examples given above.

<div class="mw-heading mw-heading4"><h4 id="Transfer_function_for_the_partial_differential_equation_example">Transfer function for the partial differential equation example</h4></div>

Setting the initial condition <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle w_{0}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle w_{0}}</annotation>

</semantics>

</math><img src="./7aa052386ec49846179aa8bbe2b279b57a675e00.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.718ex; height:2.009ex;" alt="{\displaystyle w_{0}}" loading="lazy"> equal to zero and denoting Laplace transforms with respect to t by capital letters we obtain from the partial differential equation given above

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle sW(s,\xi )=-{\frac {d}{d\xi }}W(s,\xi )+U(s),}</annotation>

</semantics>

</math><img src="./72dee3996d4db27a6064f9f83fc45dcb5878bc99.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:32.047ex; height:5.843ex;" alt="{\displaystyle sW(s,\xi )=-{\frac {d}{d\xi }}W(s,\xi )+U(s),}" loading="lazy"></dd>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle W(s,0)=0,}</annotation>

</semantics>

</math><img src="./7d4e3a57ab375564424e96d73a12de8e8818f283.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.439ex; height:2.843ex;" alt="{\displaystyle W(s,0)=0,}" loading="lazy"></dd>

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Y(s)=\int _{0}^{1}W(s,\xi )\,d\xi .}</annotation>

</semantics>

</math><img src="./4c249a9e72044d05528809967ae3e8e3204a2513.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:22.356ex; height:6.176ex;" alt="{\displaystyle Y(s)=\int _{0}^{1}W(s,\xi )\,d\xi .}" loading="lazy"></dd></dl>

This is an inhomogeneous linear differential equation with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \xi }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \xi }</annotation>

</semantics>

</math><img src="./e0b461aaf61091abd5d2c808931c48b8ff9647db.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.03ex; height:2.509ex;" alt="{\displaystyle \xi }" loading="lazy"> as the variable, s as a parameter and initial condition zero. The solution is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W(s,\xi )=U(s)(1-e^{-s\xi })/s}">

<msup>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle W(s,\xi )=U(s)(1-e^{-s\xi })/s}</annotation>

</semantics>

</math><img src="./f7a502c71878c05ae0f8c0fd4f668e5984c84765.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.339ex; height:3.176ex;" alt="{\displaystyle W(s,\xi )=U(s)(1-e^{-s\xi })/s}" loading="lazy">. Substituting this in the equation for Y and integrating gives <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y(s)=U(s)(e^{-s}+s-1)/s^{2}}">

<msup>

</mrow>

</msup>

</mrow>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Y(s)=U(s)(e^{-s}+s-1)/s^{2}}</annotation>

</semantics>

</math><img src="./14841c9c05db0acfeac0214e6f4d455a0c9ae0a2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.869ex; height:3.176ex;" alt="{\displaystyle Y(s)=U(s)(e^{-s}+s-1)/s^{2}}" loading="lazy"> so that the transfer function is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (e^{-s}+s-1)/s^{2}}">

<msup>

</mrow>

</msup>

</mrow>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (e^{-s}+s-1)/s^{2}}</annotation>

</semantics>

</math><img src="./ae030f4cc0d3e7ba4eb0fda89d3c0b49a5ee74f9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.416ex; height:3.176ex;" alt="{\displaystyle (e^{-s}+s-1)/s^{2}}" loading="lazy">.

<div class="mw-heading mw-heading4"><h4 id="Transfer_function_for_the_delay_differential_equation_example">Transfer function for the delay differential equation example</h4></div>

Proceeding similarly as for the partial differential equation example, the transfer function for the delay equation example is<a href="#cite_note-7">[7]</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1/(s-1-e^{-s})}">

</mrow>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 1/(s-1-e^{-s})}</annotation>

</semantics>

</math><img src="./fb4ea50e8452e5c4f52a6dc3b45dbcd8c07c118d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.433ex; height:3.009ex;" alt="{\displaystyle 1/(s-1-e^{-s})}" loading="lazy">.

<div class="mw-heading mw-heading3"><h3 id="Controllability">Controllability</h3></div>

In the infinite-dimensional case there are several non-equivalent definitions of <a href="Controllability" title="Controllability">controllability</a> which for the finite-dimensional case collapse to the one usual notion of controllability. The three most important controllability concepts are:

<ul><li>Exact controllability,</li>

<li>Approximate controllability,</li>

<li>Null controllability.</li></ul>

<div class="mw-heading mw-heading4"><h4 id="Controllability_in_discrete-time">Controllability in discrete-time</h4></div>

An important role is played by the maps <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Phi _{n}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Phi _{n}}</annotation>

</semantics>

</math><img src="./a4c3c96a91205fb1ae9d97b9e93b763b424bbac5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.897ex; height:2.509ex;" alt="{\displaystyle \Phi _{n}}" loading="lazy"> which map the set of all U valued sequences into X and are given by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Phi _{n}u=\sum _{k=0}^{n}A^{k}Bu_{k}}">

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</munderover>

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Phi _{n}u=\sum _{k=0}^{n}A^{k}Bu_{k}}</annotation>

</semantics>

</math><img src="./c48f163f945a14a4c6a27f562d762e6762cabc08.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:18.081ex; height:7.009ex;" alt="{\displaystyle \Phi _{n}u=\sum _{k=0}^{n}A^{k}Bu_{k}}" loading="lazy">. The interpretation is that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Phi _{n}u}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Phi _{n}u}</annotation>

</semantics>

</math><img src="./66095639aa6c0475a8358ced2eb45f9344881dc6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.226ex; height:2.509ex;" alt="{\displaystyle \Phi _{n}u}" loading="lazy"> is the state that is reached by applying the input sequence u when the initial condition is zero. The system is called

<ul><li>exactly controllable in time n if the range of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Phi _{n}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Phi _{n}}</annotation>

</semantics>

</math><img src="./a4c3c96a91205fb1ae9d97b9e93b763b424bbac5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.897ex; height:2.509ex;" alt="{\displaystyle \Phi _{n}}" loading="lazy"> equals X,</li>

<li>approximately controllable in time n if the range of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Phi _{n}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Phi _{n}}</annotation>

</semantics>

</math><img src="./a4c3c96a91205fb1ae9d97b9e93b763b424bbac5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.897ex; height:2.509ex;" alt="{\displaystyle \Phi _{n}}" loading="lazy"> is dense in X,</li>

<li>null controllable in time n if the range of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Phi _{n}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Phi _{n}}</annotation>

</semantics>

</math><img src="./a4c3c96a91205fb1ae9d97b9e93b763b424bbac5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.897ex; height:2.509ex;" alt="{\displaystyle \Phi _{n}}" loading="lazy"> includes the range of An.</li></ul>

<div class="mw-heading mw-heading4"><h4 id="Controllability_in_continuous-time">Controllability in continuous-time</h4></div>

In controllability of continuous-time systems the map <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Phi _{t}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Phi _{t}}</annotation>

</semantics>

</math><img src="./6573886a4aa9f86bb43953426774b27682cf3e12.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.504ex; height:2.509ex;" alt="{\displaystyle \Phi _{t}}" loading="lazy"> given by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{0}^{t}{\rm {e}}^{As}Bu(s)\,ds}">

</mrow>

</mrow>

</msubsup>

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \int _{0}^{t}{\rm {e}}^{As}Bu(s)\,ds}</annotation>

</semantics>

</math><img src="./a3425afe0b13089934308e3875c4428ae93a4c15.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:15.632ex; height:6.176ex;" alt="{\displaystyle \int _{0}^{t}{\rm {e}}^{As}Bu(s)\,ds}" loading="lazy"> plays the role that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Phi _{n}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Phi _{n}}</annotation>

</semantics>

</math><img src="./a4c3c96a91205fb1ae9d97b9e93b763b424bbac5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.897ex; height:2.509ex;" alt="{\displaystyle \Phi _{n}}" loading="lazy"> plays in discrete-time. However, the space of control functions on which this operator acts now influences the definition. The usual choice is L2(0, ∞;U), the space of (equivalence classes of) U-valued square integrable functions on the interval (0, ∞), but other choices such as L1(0, ∞;U) are possible. The different controllability notions can be defined once the domain of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Phi _{t}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Phi _{t}}</annotation>

</semantics>

</math><img src="./6573886a4aa9f86bb43953426774b27682cf3e12.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.504ex; height:2.509ex;" alt="{\displaystyle \Phi _{t}}" loading="lazy"> is chosen. The system is called<a href="#cite_note-8">[8]</a>

<ul><li>exactly controllable in time t if the range of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Phi _{t}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Phi _{t}}</annotation>

</semantics>

</math><img src="./6573886a4aa9f86bb43953426774b27682cf3e12.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.504ex; height:2.509ex;" alt="{\displaystyle \Phi _{t}}" loading="lazy"> equals X,</li>

<li>approximately controllable in time t if the range of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Phi _{t}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Phi _{t}}</annotation>

</semantics>

</math><img src="./6573886a4aa9f86bb43953426774b27682cf3e12.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.504ex; height:2.509ex;" alt="{\displaystyle \Phi _{t}}" loading="lazy"> is dense in X,</li>

<li>null controllable in time t if the range of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Phi _{t}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Phi _{t}}</annotation>

</semantics>

</math><img src="./6573886a4aa9f86bb43953426774b27682cf3e12.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.504ex; height:2.509ex;" alt="{\displaystyle \Phi _{t}}" loading="lazy"> includes the range of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\rm {e}}^{At}}">

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\rm {e}}^{At}}</annotation>

</semantics>

</math><img src="./a8c892785dece4c74c225e12255cb46022bce918.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.091ex; height:2.676ex;" alt="{\displaystyle {\rm {e}}^{At}}" loading="lazy">.</li></ul>

<div class="mw-heading mw-heading3"><h3 id="Observability">Observability</h3></div>

As in the finite-dimensional case, <a href="Observability" title="Observability">observability</a> is the dual notion of controllability. In the infinite-dimensional case there are several different notions of observability which in the finite-dimensional case coincide. The three most important ones are:

<ul><li>Exact observability (also known as continuous observability),</li>

<li>Approximate observability,</li>

<li>Final state observability.</li></ul>

<div class="mw-heading mw-heading4"><h4 id="Observability_in_discrete-time">Observability in discrete-time</h4></div>

An important role is played by the maps <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Psi _{n}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Psi _{n}}</annotation>

</semantics>

</math><img src="./9e979a931bf82db45f2c63353f9bc8bbbbee102b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.027ex; height:2.509ex;" alt="{\displaystyle \Psi _{n}}" loading="lazy"> which map X into the space of all Y valued sequences and are given by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\Psi _{n}x)_{k}=CA^{k}x}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (\Psi _{n}x)_{k}=CA^{k}x}</annotation>

</semantics>

</math><img src="./b0c143a508de60f601de274781f99671f4b26b19.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.281ex; height:3.176ex;" alt="{\displaystyle (\Psi _{n}x)_{k}=CA^{k}x}" loading="lazy"> if k ≤ n and zero if k > n. The interpretation is that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Psi _{n}x}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Psi _{n}x}</annotation>

</semantics>

</math><img src="./be189404675bfc8d79e50ca2a737addea7b640c2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.356ex; height:2.509ex;" alt="{\displaystyle \Psi _{n}x}" loading="lazy"> is the truncated output with initial condition x and control zero. The system is called

<ul><li>exactly observable in time n if there exists a kn > 0 such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \|\Psi _{n}x\|\geq k_{n}\|x\|}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \|\Psi _{n}x\|\geq k_{n}\|x\|}</annotation>

</semantics>

</math><img src="./6da2a4466302048772efd0b59e357f45f974b72b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.864ex; height:2.843ex;" alt="{\displaystyle \|\Psi _{n}x\|\geq k_{n}\|x\|}" loading="lazy"> for all x ∈ X,</li>

<li>approximately observable in time n if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Psi _{n}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Psi _{n}}</annotation>

</semantics>

</math><img src="./9e979a931bf82db45f2c63353f9bc8bbbbee102b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.027ex; height:2.509ex;" alt="{\displaystyle \Psi _{n}}" loading="lazy"> is <a href="Injective" class="mw-redirect" title="Injective">injective</a>,</li>

<li>final state observable in time n if there exists a kn > 0 such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \|\Psi _{n}x\|\geq k_{n}\|A^{n}x\|}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \|\Psi _{n}x\|\geq k_{n}\|A^{n}x\|}</annotation>

</semantics>

</math><img src="./7c5d30a4a96d41595bf45ba69b6fd19b011ba934.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.826ex; height:2.843ex;" alt="{\displaystyle \|\Psi _{n}x\|\geq k_{n}\|A^{n}x\|}" loading="lazy"> for all x ∈ X.</li></ul>

<div class="mw-heading mw-heading4"><h4 id="Observability_in_continuous-time">Observability in continuous-time</h4></div>

In observability of continuous-time systems the map <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Psi _{t}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Psi _{t}}</annotation>

</semantics>

</math><img src="./5a87f0d64f2e13593293205ba5daa798d55ec592.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.634ex; height:2.509ex;" alt="{\displaystyle \Psi _{t}}" loading="lazy"> given by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\Psi _{t})(s)=C{\rm {e}}^{As}x}">

<msub>

</mrow>

</msub>

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (\Psi _{t})(s)=C{\rm {e}}^{As}x}</annotation>

</semantics>

</math><img src="./64c1e400555efde8dddb8dc822d8e41a7fbcaae0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.806ex; height:3.176ex;" alt="{\displaystyle (\Psi _{t})(s)=C{\rm {e}}^{As}x}" loading="lazy"> for s∈[0,t] and zero for s>t plays the role that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Psi _{n}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Psi _{n}}</annotation>

</semantics>

</math><img src="./9e979a931bf82db45f2c63353f9bc8bbbbee102b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.027ex; height:2.509ex;" alt="{\displaystyle \Psi _{n}}" loading="lazy"> plays in discrete-time. However, the space of functions to which this operator maps now influences the definition. The usual choice is L2(0, ∞, Y), the space of (equivalence classes of) Y-valued square integrable functions on the interval (0,∞), but other choices such as L1(0, ∞, Y) are possible. The different observability notions can be defined once the co-domain of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Psi _{t}}">

<msub>

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<annotation encoding="application/x-tex">{\displaystyle \Psi _{t}}</annotation>

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</math><img src="./5a87f0d64f2e13593293205ba5daa798d55ec592.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.634ex; height:2.509ex;" alt="{\displaystyle \Psi _{t}}" loading="lazy"> is chosen. The system is called<a href="#cite_note-9">[9]</a>

<ul><li>exactly observable in time t if there exists a kt > 0 such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \|\Psi _{t}x\|\geq k_{t}\|x\|}">

<msub>

</mrow>

</msub>

<msub>

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<annotation encoding="application/x-tex">{\displaystyle \|\Psi _{t}x\|\geq k_{t}\|x\|}</annotation>

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</math><img src="./70e67114881480f84e1ea02915761824ef0a9e98.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.079ex; height:2.843ex;" alt="{\displaystyle \|\Psi _{t}x\|\geq k_{t}\|x\|}" loading="lazy"> for all x ∈ X,</li>

<li>approximately observable in time t if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Psi _{t}}">

<msub>

</mrow>

</msub>

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<annotation encoding="application/x-tex">{\displaystyle \Psi _{t}}</annotation>

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</math><img src="./5a87f0d64f2e13593293205ba5daa798d55ec592.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.634ex; height:2.509ex;" alt="{\displaystyle \Psi _{t}}" loading="lazy"> is <a href="Injective" class="mw-redirect" title="Injective">injective</a>,</li>

<li>final state observable in time t if there exists a kt > 0 such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \|\Psi _{t}x\|\geq k_{t}\|{\rm {e}}^{At}x\|}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msup>

</mrow>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle \|\Psi _{t}x\|\geq k_{t}\|{\rm {e}}^{At}x\|}</annotation>

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</math><img src="./3b5211cfcae05286697d53f60ae4cc405125e0fb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.17ex; height:3.176ex;" alt="{\displaystyle \|\Psi _{t}x\|\geq k_{t}\|{\rm {e}}^{At}x\|}" loading="lazy"> for all x ∈ X.</li></ul>

<div class="mw-heading mw-heading3"><h3 id="Duality_between_controllability_and_observability">Duality between controllability and observability</h3></div>

As in the finite-dimensional case, controllability and observability are dual concepts (at least when for the domain of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Phi }">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle \Phi }</annotation>

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</math><img src="./aed80a2011a3912b028ba32a52dfa57165455f24.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \Phi }" loading="lazy"> and the co-domain of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Psi }">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle \Psi }</annotation>

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</math><img src="./f5471531a3fe80741a839bc98d49fae862a6439a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \Psi }" loading="lazy"> the usual L2 choice is made). The correspondence under duality of the different concepts is:<a href="#cite_note-10">[10]</a>

<ul><li>Exact controllability ↔ Exact observability,</li>

<li>Approximate controllability ↔ Approximate observability,</li>

<li>Null controllability ↔ Final state observability.</li></ul>

<ul><li><a href="Control_theory" title="Control theory">Control theory</a></li>

<li><a href="State_space_(controls)" class="mw-redirect" title="State space (controls)">State space (controls)</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2></div>

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</style><div class="reflist reflist-columns references-column-width reflist-columns-2">

<li id="cite_note-1"><a href="#cite_ref-1">^</a> Curtain and Zwart

</li>

<li id="cite_note-2"><a href="#cite_ref-2">^</a> Curtain and Zwart Example 2.2.4

</li>

<li id="cite_note-3"><a href="#cite_ref-3">^</a> Curtain and Zwart Theorem 2.4.6

</li>

<li id="cite_note-4"><a href="#cite_ref-4">^</a> This is the mathematical convention, engineers seem to prefer transfer functions to be holomorphic at infinity; this is achieved by replacing z by 1/z

</li>

<li id="cite_note-5"><a href="#cite_ref-5">^</a> Curtain and Zwart Lemma 4.3.6

</li>

<li id="cite_note-6"><a href="#cite_ref-6">^</a> Staffans Theorem 4.6.7

</li>

<li id="cite_note-7"><a href="#cite_ref-7">^</a> Curtain and Zwart Example 4.3.13

</li>

<li id="cite_note-8"><a href="#cite_ref-8">^</a> Tucsnak Definition 11.1.1

</li>

<li id="cite_note-9"><a href="#cite_ref-9">^</a> Tucsnak Definition 6.1.1

</li>

<li id="cite_note-10"><a href="#cite_ref-10">^</a> Tucsnak Theorem 11.2.1

</li>

</ol></div>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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</style><cite id="CITEREFCurtainZwart1995" class="citation cs2"><a href="Ruth_F._Curtain" title="Ruth F. Curtain">Curtain, Ruth</a>; Zwart, Hans (1995), An Introduction to Infinite-Dimensional Linear Systems theory, Springer</cite></li>

<li><cite id="CITEREFTucsnakWeiss2009" class="citation cs2">Tucsnak, Marius; Weiss, George (2009), Observation and Control for Operator Semigroups, Birkhauser</cite></li>

<li><cite id="CITEREFStaffans2005" class="citation cs2">Staffans, Olof (2005), Well-posed linear systems, Cambridge University Press</cite></li>

<li><cite id="CITEREFLuoGuoMorgul1999" class="citation cs2">Luo, Zheng-Hua; Guo, Bao-Zhu; Morgul, Omer (1999), Stability and Stabilization of Infinite Dimensional Systems with Applications, Springer</cite></li>

<li><cite id="CITEREFLasieckaTriggiani2000" class="citation cs2"><a href="Irena_Lasiecka" title="Irena Lasiecka">Lasiecka, Irena</a>; Triggiani, Roberto (2000), Control Theory for Partial Differential Equations, Cambridge University Press</cite></li>

<li><cite id="CITEREFBensoussanDa_PratoDelfourMitter2007" class="citation cs2">Bensoussan, Alain; Da Prato, Giuseppe; Delfour, Michel; Mitter, Sanjoy (2007), Representation and Control of Infinite Dimensional Systems (second ed.), Birkhauser</cite></li></ul>

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