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.


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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[1] Box, G.; Draper, N., Empirical model building and response surfaces, (1987), Wiley New York · Zbl 0614.62104
[2] Ji, G.; Lasiecka, I., Partially observed analytic systems with fully unbounded actuators and sensors - FEM algorithms, Comput. optim. appl., 11, 111, (1998) · Zbl 0914.93060
[3] G. Ji, T.E. Peterson, Continuous time Galerkin methods for parabolic equations with time-dependent coefficients, Int. J. Appl. Math. Comput. Sci., to appear
[4] Krishnan, K.; Brodeur, J., Toxicological consequences of combined exposure to environmental pollutants, Aces, 3, 3, 1, (1991)
[5] Lasiecka, I.; Triggiani, R., Dirichlet boundary control problem for parabolic equations with quadratic cost: analyticity and Riccati’s feedback synthesis, SIAM J. control optim., 21, 41, (1983) · Zbl 0506.93036
[6] Lasiecka, I.; Triggiani, R., Riccati differential equations with unbounded coefficients and nonsmooth terminal condition – the case of analytic semigroups, SIAM J. math. anal., 23, 449, (1992) · Zbl 0774.34047
[7] I. Lasiecka, R. Triggiani, Differential and algebraic Riccati equations with application to boundary/point control problems: continuous theory and approximation theory, Lecture Notes in Control and Information Sciences, vol. 164, Springer, Berlin, 1991 · Zbl 0754.93038
[8] C. Martin, M. Egerstedt, S. Sun, Optimal control, statistics and path planning, Math. Comput. Modeling, to appear · Zbl 0976.65057
[9] Montgomery, D., Design and analysis of experiments, (1991), Wiley New York · Zbl 0747.62072
[10] Pazy, A., Semigroups of operators and applications to partial differential equations, (1983), Springer New York · Zbl 0516.47023
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