We study optimal control problems for partial differential equations (focusing on the multidimensional differential equation) with control functions in the Dirichlet boundary conditions under pointwise control (and we admit state - by assuming weak hypotheses) constraints.
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We discuss the solvability of a nonhomogeneous boundary value problem for the semilinear equation of the vibrating string $x_{tt}(t,y) - Δx(t,y) + f(t,y,x(t,y)) = 0$ in a bounded domain and with a certain type of superlinear nonlinearity. To this end we derive a new dual variational method.
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