An Optimization Method for Semilinear Parabolic Relaxed Constrained Optimal Control Problems

Abstract

This paper addresses optimal control problems governed by semilinear parabolic partial differential equations, subject to control constraints and state constraints of integral type. Since such problems may not have classical solutions, a relaxed optimal control problem is considered. The relaxed control problem is discretized by using a finite element method and the behavior in the limit of discrete optimality, admissibility and extremality properties is studied. A conditional descent method with penalties applied to the discrete problems is proposed. It is shown that the accumulation points of sequences produced by this method are admissible and extremal for the discrete problem. Finally, numerical examples are given.

Keywords

• Conditional descent method,Discretization,Optimal control,Relaxed controls,Semilinear parabolic equations,State constraints

References

1. [1] J. Warga, Optimal Control of Differential and Functional Equations, Academic Press, New York, 1972.
2. [2] T. Roubiček, Relaxation in Optimization Theory and Variational Calculus, Walter de Gruyter, Berlin, 1997.
3. [3] H. O. Fattorini, Infinite Dimensional Optimization Theory and Optimal Control, Cambridge Univ. Press, Cambridge, 1999.
4. [4] J. Warga, Steepest descent with relaxed controls, SIAM J. Control Optim., 15 (1977), 674-682.
5. [5] I. Chryssoverghi, A. Bacopoulos, B. Kokkinis, J. Coletsos, Mixed Frank-Wolfe penalty method with applications to nonconvex optimal control problems, J. Optimiz. Theory App., 94 (1997) 311-334.
6. [6] I. Chryssoverghi, A. Bacopoulos, Approximation of relaxed nonlinear parabolic optimal control problems, J. Optimiz. Theory App., 77 (1993) 31-50.
7. [7] T. Roubiček, A convergent computational method for constrained optimal relaxed control problems, J. Optimiz. Theory App., 69 (1991) 589-603.
8. [8] V. Azhmyakov, W. Schmidt, Approximations of relaxed optimal control problems, J. Optimiz. Theory App., 130 (2006) 61-77.

9. [9] N. Arada, J. P. Raymond, State-constrained relaxed problems for semilinear elliptic equations, J. Math. Anal. Appl., 223 (1998) 248-271.
10. [10] N. Arada, J. P. Raymond, Approximation of optimal control problems with state constraints, Numer. Funct. Anal. and Optimiz., 21 (2000) 601-621.
11. [11] E. Casas, The relaxation theory applied to optimal control problems of semilinear elliptic equations, J. Dolezal, J. Fidler (editors), System Modelling and Optimization, Chapman & Hall, London, 1996, pp. 187-194.
12. [12] I. Chryssoverghi, J. Coletsos, B. Kokkinis, Discrete relaxed method for semilinear parabolic optimal control problems, Control Cybernet., 28 (1999) 157-176.
13. [13] S. Luan, Nonexistence and existence of an optimal control problem governed by a class of semilinear elliptic equations, J. Optimiz. Theory App., 158 (2013) 1-10.
14. [14] I. Chryssoverghi, J. Coletsos, B. Kokkinis, Classical and relaxed optimization methods for nonlinear parabolic optimal control problems I. Lirkov, S. Margenov and J. Wasniewski (editors), Large-scale scientific computing, Springer-Verlag, Berlin, 2010, pp. 247-255.
15. [15] J. L. Lions, Quelques M´ethodes de R´esolution des Prob`emes aux Limites Non Lin´eaires, Dunod Gauthier-Villars, Paris, 1969.
16. [16] R. Temam, Navier-Stokes Equations, North-Holland, New York, 1977.
17. [17] S. Treanta, On signomial constrained optimal control problems, Commun. Adv. Math. Sci., 2 (2019), 55-59.