Existence, Uniqueness and Convergence of Simultaneous Distributed-Boundary Optimal Control Problems

• Autorzy: Claudia M. Gariboldi , Domingo A. Tarzia
• Języki publikacji: EN

Abstrakty

EN

We consider a steady-state heat conduction problem P for the Poisson equation with Mied Bondary conditions in a bounded multidimensional domain Ω. We also consider a family of problems P_α for the same Poisson equation with mixed boundary conditions, α > 0 being the heat transfer coefficient defined on a portion Γ₁ of the boundary. We formulate simultaneous distributed and Neumann boundary optimal control problems on the internal energy g within Ω and the heat flux q, defined on the complementary portion Γ₂ of the boundary of Ω for quadratic cost functional. Here, the control variable is the vector (g,q). We prove existence and uniqueness of the optimal control (g,q) for the system state of P, and (gα,qα) for the system state of P_α, for each α > 0, and we give the corresponding optimality conditions. We prove strong convergence, in suitable Sobolev spaces, of the vectorial optimal controls, system and adjoint states governed by the problems P_α to the corresponding vectorial optimal control, system and adjoint states governed by the problem P, when the parameter α goes to infinity. We also obtain estimations between the solutions of these vectorial optimal control problems and the solution of two scalar optimal control problems characterized by fixed g (with boundary optimal control q) and fixed q (with distributed optimal control g), respectively, for cases both of α > 0 and α = ∞. (original abstract)

Słowa kluczowe

EN

Control theory; Optimization theory

PL

Teoria kontroli; Teoria optymalizacji

• Czasopismo: Control and Cybernetics
• Rocznik: 2015
• Tom: 44
• Numer: nr 1
• Strony: 5--17

Twórcy

autor

Claudia M. Gariboldi

• Universidad Nacional de Río Cuarto, Argentina

autor

Domingo A. Tarzia

• Universidad Austral, Paraguay

Bibliografia

• BEN BELGACEM, F., EL FEKIH, H., METOUI, H. (2003) Singular perturbation for the Dirichlet boundary control of elliptic problems. ESAIM: M2AN 37, 833-850.
• BENSOUSSAN, A. (1974) Teor´ıa moderna de control óptimo. Cuadern. Inst. Mat. Beppo Levi