Dirichlet boundary control of semilinear parabolic equations. II: Problems with pointwise state constraints
The systems are described by the parabolic equation \[ {\partial y(t, x) \over \partial t} + Ay(t, x) + \Phi(t, x, y(t, x), u(t, x)) = 0 \] in a cylinder \((0, T) \times \Omega,\) \(\Omega\) an \(n\)-dimensional domain with boundary \(\Gamma,\) with an initial condition on the base of the cylinder and the Dirichlet boundary condition \(y(t, x) = v(t, x)\) on \((0, T) \times \Gamma;\) \(A\) is a second order elliptic operator. The optimal problem is that of minimizing an integral cost functional under a pointwise state constraint. In order to obtain a version of Pontryagin's maximum principle for this kind of problem one needs to pick multipliers in the dual of \(L^\infty((0, T) \times \Omega).\) These multipliers are finitely additive measures and produce an adjoint equation not interpretable in the sense of distributions. The authors tackle this problem by proving a theorem on decomposition of finitely additive measures into a regular part (a Radon measure) and a finitely additive part. Only the regular measure intervenes in the adjoint equation; the other measure plays a role in the part of Pontryagin's principle involving the boundary control \(v(t, x).\) [Cf. also Part I in Appl. Math. Optimization 45, No.~2, 125--143 (2002; Zbl 1005.49016)].
scientific article; zbMATH DE number 1753056