FWF Project P18056-N12
Runtime: 01.09.2005-31.08.2008

Project Team

• Nataliya Metla
• Arnd Rösch (project leader)
• Roland Griesse (project leader)

Project Abstract

Many technical processes are described by partial differential equations. The optimization of such processes or identification of material parameters leads to optimal control problems for partial differential equations. Naturally, some quantities of the process have to be restricted to admissible ranges. The scope of this project covers optimal control of elliptic and parabolic partial differential equations with pointwise inequality constraints in space and time.

Typically, nonlinear functions are involved in real-life problems. In turn, necessary and sufficient optimality conditions of nonlinear optimal control problems contain first and second derivatives of these nonlinearities. Sufficient optimality conditions can ensure stability under perturbations of the solutions of the investigated optimal control problems. Moreover, they represent the key to prove convergence of fast and efficient numerical methods.

Until now, sufficient optimality conditions, stability results, and convergence of fast numerical methods are only known in case the pointwise inequality constraints affect solely the controls of the system. In contrast, real-life problems contain typically both, pointwise inequality constraints for controls and process quantities, i.e., states. Inequality constraints for process quantities alone lead to mathematical problems which are far from being solved.

In this project, we will establish sufficient optimality conditions and we will prove stability results and convergence of the SQP-method for mixed constrained optimal control problems: Pointwise inequality conditions containing controls and process quantities are simultaneously involved in such constraints. These theory developed in this project will guarantee reliable numerical results for arbitrary fine discretizations of the involved partial differential equations.

Keywords and AMS Classification

• optimal control
• convergence theory
• partial differential equations
• sufficient optimality conditions
• SQP method
• mixed constraints

AMS Subject Classification: 49K20, 49K40, 49M05, 49M37

Related Publications

• R. Griesse, N. Metla and A. Rösch
  Local Quadratic Convergence of SQP for Elliptic Optimal Control Problems with Nonlinear Mixed Control-State Constraints
  submitted, 2008
• R. Griesse, N. Metla and A. Rösch
  Local Quadratic Convergence of SQP for Elliptic Optimal Control Problems with Mixed Control-State Constraints
  submitted, 2007
• W. Alt, R. Griesse, N. Metla and A. Rösch
  Lipschitz Stability for Elliptic Optimal Control Problems with Mixed Control-State Constraints
  submitted, 2006
• R. Griesse
  Lipschitz Stability of Solutions to Some State-Constrained Elliptic Optimal Control Problems
  Journal of Analysis and its Applications (ZAA), 25, p.435-455, 2006
• A. Rösch and F. Tröltzsch
  Existence of Regular Lagrange Multipliers for a Nonlinear Elliptic Optimal Control Problem with Pointwise Control-State Constraints
  SIAM Journal on Control and Optimization (SICON), 45(2), p.548-564, 2006
• A. Rösch and F. Tröltzsch
  Sufficient Second-Order Optimality Conditions for an Elliptic Optimal Control Problem with Pointwise Control-State Constraints
  to appear in: SIAM Journal on Optimization (SIOPT)