SUFFICIENT OPTIMALITY IN A PARABOLIC CONTROL PROBLEM

We define a class of parabolic problems with control and state constraints and identify a problem within this class which possesses a locally unique critical point satisfying the second order sufficient optimality conditions. In this solution inequality constraints on the control are strongly active. The second derivative of the Lagrangian is not globally coercive. This is both shown analytically as well as verified numerically for a finite difference discretization.

[1]  Jorge Nocedal,et al.  An Interior Point Algorithm for Large-Scale Nonlinear Programming , 1999, SIAM J. Optim..

[2]  E. Casas.  Pontryagin's Principle for State-Constrained Boundary Control Problems of Semilinear Parabolic Equations , 1997 .

[3]  Deborah Sulsky,et al.  Row Ordering for a Sparse QR Decomposition , 1994 .

[4]  Jean-Pierre Raymond,et al.  Hamiltonian Pontryagin's Principles for Control Problems Governed by Semilinear Parabolic Equations , 1999 .

[5]  Fredi Tröltzsch,et al.  Second Order Sufficient Optimality Conditions for Nonlinear Parabolic Control Problems with State Constraints , 2000 .

[6]  Peter Spellucci,et al.  Numerische Verfahren der nichtlinearen Optimierung , 1993 .

[7]  Fredi Tröltzsch,et al.  Second Order Sufficient Optimality Conditions for Some State-constrained Control Problems of Semilinear Elliptic Equations , 2000, SIAM J. Control. Optim..

[8]  Robert J. Vanderbei,et al.  An Interior-Point Algorithm for Nonconvex Nonlinear Programming , 1999, Comput. Optim. Appl..

[9]  Brian W. Kernighan,et al.  AMPL: A Modeling Language for Mathematical Programming , 1993 .

[10]  Fredi Tröltzsch,et al.  On an Augmented Lagrangian SQP Method for a Class of Optimal Control Problems in Banach Spaces , 2002, Comput. Optim. Appl..

[11]  Fredi Tröltzsch,et al.  Second-Order Necessary and Sufficient Optimality Conditions for Optimization Problems and Applications to Control Theory , 2002, SIAM J. Optim..

[12]  Hans D. Mittelmann.  Verification of Second-Order Sufficient Optimality Conditions for Semilinear Elliptic and Parabolic Control Problems , 2001, Comput. Optim. Appl..