Optimal control of parabolic equation with an infinite number of variables for non-standard functional and time delay

A distributed control problem for the parabolic operator withan infinite number of variables and time delay is considered.The performance index has an integral form. Constraints on controlsare imposed. To obtain optimality conditions for the Neumannproblem, the generalization of the Dubovitskii–Milyutintheorem given by Walczak in WALCZAK, S. Folia Mathematics, 1,187–196 and WALCZAK, S. J. Optim. Theory Appl., 42, 561–582was applied.     

Optimal control of a system governed by a parabolic equation with an infinite number of variables and time delay

A distributed control problem with delay for the parabolic operator with an infinite number of variables is considered. The performance index has an integral form. Constraints on controls are assumed. To obtain optimality conditions, the generalization of the Dubovicki-Milutin theorem given by Walczak in Ref. 1 was applied.     

Quadratic Pareto optimal control of parabolic equation with state-control constraints and an infinite number of variables

A distributed Pareto optimal control problem for the parabolicoperator with an infinite number of variables is considered.The performance index has an integral form. Constraints on controlsand on states are imposed. To obtain optimality conditions forthe Neumann problem, the generalization of the Dubovitskii–MilyutinTheorem given by WALCZAK, S. (1984a) Folia Mathematica, 1, 187–196and (1984b) J. Optimiz. Theory Appl., 42, 561–582, wasapplied.     

Time-optimal control problem for parabolic equations with control constraints and infinite number of variables

A time optimal control problem for parabolic equations withan infinite number of variables is considered. A time optimalcontrol problem is replaced by an equivalent one with a performanceindex in the form of integral form. Constraints on controlsare assumed. To obtain the optimality conditions for the Neumannproblem, the generalization of the Dubovitskii–Milyutintheorem given by Walczak (1984, Acta Universitatis LodziensisFolia Mathematica, 187–196; 1984, J. Optim. Theor. Appl.,42, 561–582) was applied.