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Zbl 0893.49017
Casas, Eduardo
Pontryagin's principle for state-constrained boundary control problems of semilinear parabolic equations.
(English)
[J] SIAM J. Control Optimization 35, No. 4, 1297-1327 (1997). ISSN 0363-0129; ISSN 1095-7138/e

The state equation is $${\partial \over \partial t} y(t, x) + Ay(t, x) + a_0(t, x, y(t, x)) = 0$$ where $A$ is a linear elliptic differential operator in a domain $\Omega \subset{\bbfR}^m$ with boundary $\Gamma.$ The (controlled) boundary condition is $$\partial_{\nu_A} y(t, x) = f(t, x, y(t, x), u(t, x))$$ where $\partial_{\nu_A}$ is a directional derivative on $\Gamma$ associated with $A.$ The problem includes constraints on the control $u(t, x)$ as well as on the state $y(t, x),$ and the cost functional to be minimized in a time interval $0 \le t \le T$ is $$J(u) = \int_{(0, T) \times \Omega} L(t, x, y_u(t, x)) dt dx + \int_{(0, T) \times \Sigma} l(t, x, y_u(t, x), u(t, x)) dt d\sigma(x).$$ The author derives a maximum principle of Pontryagin's type for controls minimizing the cost functional under all constraints. Besides the new result, the paper is a very good survey of the different approaches to this sort of problems (for instance, spike vs. patch or diffuse perturbations) and of the difficulties associated with the adjoint variational equation, whose inhomogeneous term is a measure rather than a function.
[H.O.Fattorini]
MSC 2000:
*49K20 Optimal control problems with PDE (nec./ suff.)
35J20 Second order elliptic equations, variational methods
93C20 Control systems governed by PDE

Keywords: Pontryagin principle; boundary control; semilinear parabolic equations; optimality conditions; state constraints

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