##
**An optimal control problem for a parabolic equation in non-cylindrical domains.**
*(English)*
Zbl 0656.49001

The authors study an optimal control problem for a parabolic equation with homogeneous Dirichlet data described by
\[
(1)\quad (\partial /\partial t)y(t,x)=\Delta y(t,x)+u(t,x),\quad t\geq 0,\quad x\in \Omega_ t,
\]

\[ y(0,x)=y_ 0(x),\quad x\in \Omega_ 0,\quad y(t,x)=0,\quad t\geq 0,\quad x\in \Gamma_ t. \] Here, \(\Omega_ t\) is a bounded open set of \({\mathbb{R}}^ n\) with smooth boundary \(\Gamma_ t\); y is the state, and u the control belonging to \(L^ 2([0,\infty)\times \Omega_ t)\). The problem is to find a u minimizing the cost \[ J(u)=\int^{\infty}_{0}dt\int_{\Omega_ t}(| y|^ 2+| u|^ 2)dx. \] The point here is that equation (1) is written in the non-cylindrical domain \(U_{t>0}(t\times \Omega_ t)\). Equation (1) is, however, transformed into a standard equation in a cylindrical domain by change of variables: A system of ordinary differential equations is considered: \[ (2)\quad d\beta /dt=V(t,\beta),\quad \beta (0)=x\in {\mathbb{R}}^ n. \] An assumption on the function V is that the solution to (2), denoted by \(T_ t(x)\), satisfies \(T_ t(\Omega_ 0)=\Omega_ t\) and \(T_ t(\Gamma_ 0)=\Gamma_ t\), \(t\geq 0\). By setting \(z(t,X)=y(t,T_ t(X))\) and \(v(t,X)=u(t,T_ t(X))\), equation (1) is transformed into an equation in \([0,\infty)\times \Omega_ 0\), or an abstract equation in \(L^ 2(\Omega_ 0)\); \[ (3)\quad dz/dt=B(t)z+v,\quad z(0)=y_ 0. \] Here, B(t) is derived from \(\Delta\), and \(D(B(t))=H^ 2(\Omega_ 0)\cap H^ 1_ 0(\Omega_ 0)\). Then, the original problem is equivalent to finding a v minimizing \[ \bar J(v)=\int^{\infty}_{0}dt\int_{\Omega_ 0}| \det DT_ t| (| z|^ 2+| v|^ 2)dx, \] \(DT_ t\) being the Jacobian matrix of \(T_ t\). The other assumption is the existence of a bounded domain \(\sigma\) such that \(\Omega_ t\subset \sigma\), \(t\geq 0\) in order to ensure admissible controls. At this stage, it is standard to solve the optimal control problem for equation (3). As usual, the Riccati equation associated with (3) is solved. The unique optimal control \(v'=v^*\) is found in the feedback form. The solution to the original optimal control problem is finally stated via the mapping \(T_ t\).

\[ y(0,x)=y_ 0(x),\quad x\in \Omega_ 0,\quad y(t,x)=0,\quad t\geq 0,\quad x\in \Gamma_ t. \] Here, \(\Omega_ t\) is a bounded open set of \({\mathbb{R}}^ n\) with smooth boundary \(\Gamma_ t\); y is the state, and u the control belonging to \(L^ 2([0,\infty)\times \Omega_ t)\). The problem is to find a u minimizing the cost \[ J(u)=\int^{\infty}_{0}dt\int_{\Omega_ t}(| y|^ 2+| u|^ 2)dx. \] The point here is that equation (1) is written in the non-cylindrical domain \(U_{t>0}(t\times \Omega_ t)\). Equation (1) is, however, transformed into a standard equation in a cylindrical domain by change of variables: A system of ordinary differential equations is considered: \[ (2)\quad d\beta /dt=V(t,\beta),\quad \beta (0)=x\in {\mathbb{R}}^ n. \] An assumption on the function V is that the solution to (2), denoted by \(T_ t(x)\), satisfies \(T_ t(\Omega_ 0)=\Omega_ t\) and \(T_ t(\Gamma_ 0)=\Gamma_ t\), \(t\geq 0\). By setting \(z(t,X)=y(t,T_ t(X))\) and \(v(t,X)=u(t,T_ t(X))\), equation (1) is transformed into an equation in \([0,\infty)\times \Omega_ 0\), or an abstract equation in \(L^ 2(\Omega_ 0)\); \[ (3)\quad dz/dt=B(t)z+v,\quad z(0)=y_ 0. \] Here, B(t) is derived from \(\Delta\), and \(D(B(t))=H^ 2(\Omega_ 0)\cap H^ 1_ 0(\Omega_ 0)\). Then, the original problem is equivalent to finding a v minimizing \[ \bar J(v)=\int^{\infty}_{0}dt\int_{\Omega_ 0}| \det DT_ t| (| z|^ 2+| v|^ 2)dx, \] \(DT_ t\) being the Jacobian matrix of \(T_ t\). The other assumption is the existence of a bounded domain \(\sigma\) such that \(\Omega_ t\subset \sigma\), \(t\geq 0\) in order to ensure admissible controls. At this stage, it is standard to solve the optimal control problem for equation (3). As usual, the Riccati equation associated with (3) is solved. The unique optimal control \(v'=v^*\) is found in the feedback form. The solution to the original optimal control problem is finally stated via the mapping \(T_ t\).

Reviewer: T.Nambu

### MSC:

49J20 | Existence theories for optimal control problems involving partial differential equations |

35K20 | Initial-boundary value problems for second-order parabolic equations |

93C20 | Control/observation systems governed by partial differential equations |

35B37 | PDE in connection with control problems (MSC2000) |

93C05 | Linear systems in control theory |

### Keywords:

optimal control; parabolic equation with homogeneous Dirichlet data; non- cylindrical domain; Riccati equation; feedback
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\textit{G. Da Prato} and \textit{J. P. Zolésio}, Syst. Control Lett. 11, No. 1, 73--77 (1988; Zbl 0656.49001)

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### References:

[1] | G. Da Prato and A. Ichikawa, Quadratic control for linear time varying systems, Submitted · Zbl 0692.49006 |

[2] | Lions, J.L.; Magenes, E., Problémes aux limites non homogénes et applications, () · Zbl 0165.10801 |

[3] | Zolésio, J.P., Identification de domaines par déformation, () · Zbl 0556.35129 |

[4] | Zolésio, J.P., The material derivative, (), 1089-1151 |

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