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Optimal boundary control of the heat equation with target function at terminal time. (English) Zbl 1040.49037

The paper considers the problem of minimizing the functional \[ I= \int_\Omega| y(T, x)- h(x)|^2\,dx+ \int^T_0 \int_{\partial\Omega}| u(t,x)|^2\,dS\,dt \] subject to \[ \begin{gathered} y_t=\Delta y\quad\text{in }(0,T)\times \Omega,\quad y(0,\cdot)= y_0(\cdot),\\ y(t, x)= u(t,x)\quad\text{on }(0,T)\times \partial\Omega,\end{gathered} \] where \(\Omega\subset\mathbb{R}^2\) is a Lipschitz domain with boundary \(\partial\Omega\), \(y_0\) and \(h\) are given functions. The authors reformulate the original problem as an abstract one, derive the optimality system and obtain the optimal feedback by using the formal solution of the corresponding Riccati equation.

MSC:

49N35 Optimal feedback synthesis
49J20 Existence theories for optimal control problems involving partial differential equations
35B37 PDE in connection with control problems (MSC2000)
35K05 Heat equation
93C20 Control/observation systems governed by partial differential equations
49K20 Optimality conditions for problems involving partial differential equations
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