Optimal control of parabolic differential inclusions in one space dimension

The paper is devoted to the optimisation of parabolic differential inclusions (DFIs) in a bounded rectangular region. For this, a problem with a parabolic discrete inclusion is defined, which is the main inevitable auxiliary problem. With the help of locally adjoint mappings, necessary and sufficient conditions for the optimality of parabolic discrete inclusions are proved. Then, using the method of discretization of parabolic DFIs and the already obtained optimality conditions for discrete inclusions, the necessary and sufficient conditions for the discrete-approximate problem are formulated in the form of the Euler–Lagrange type inclusion. Thus, using a specially proved equivalence theorem, we establish sufficient optimality conditions for a parabolic DFI. To demonstrate the above approach, some linear problems and polyhedral optimisation with inclusions of parabolic type are investigated. In addition, numerical results are also presented.