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.

We discuss the solvability of a nonhomogeneous boundary value problem for the semilinear equation of the vibrating string ${x}_{tt}(t,y)-\Delta x(t,y)+f(t,y,x(t,y))=0$ in a bounded domain and with a certain type of superlinear nonlinearity. To this end we derive a new dual variational method.

In this paper we establish the existence of nontrivial solutions to $$\frac{\mathrm{d}}{\mathrm{d}t}{L}_{{x}^{\text{'}}}(t,{x}^{\text{'}}\left(t\right))+{V}_{x}(t,x\left(t\right))=0,\phantom{\rule{1.0em}{0ex}}x\left(0\right)=0=x\left(T\right),$$
with ${V}_{x}$ superlinear in $x$.

The existence of positive solutions for a nonlocal boundary-value problem with vector-valued response is investigated. We develop duality and variational principles for this problem. Our variational approach enables us to approximate solutions and give a measure of a duality gap between the primal and dual functional for minimizing sequences.

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