Optimal control problems for distributed parameter systems in Banach spaces (Q688861)

The main purpose of this paper is to derive Pontryagin's maximum principle for certain semilinear parabolic distributed parameter control problems. The novelty of the approach is to study such semilinear parabolic equations in the space \(C(\overline\Omega)\), where \(\Omega\) is the bounded spatial domain in which the space variable lies. This is in contrast to the usual (and simpler) approach where the equation is treated in \(L^ p(\Omega)\), \(1< p<\infty\), because this does not enable one to handle some target conditions, or an alternative approach in \(L^ \infty(\Omega)\), because (among other reasons) the dual of \(L^ \infty(\Omega)\) which is essential for the formulation of the maximum principle is unwidely. The author starts with an abstract framework and considers first the infinite-dimensional nonlinear programming problem of minimizing a real-valued function \(f_ 0(u)\) defined in a metric space \(V\) subject to the constraint \(f(u)\in Y\), where \(f(u)\) is the defined in \(V\) and takes values in a Banach space \(E\) and \(Y\) is a subset of \(E\). The resulting Kuhn-Tucker condition then leads to the maximum principle mentioned above. The results apply to systems described by nonlinear heat equations and reaction-diffusion equations.