Abstract
Quadratic control problems for parabolic equations withstate constraints are considered. Regularity (smoothness) of the optimal solution is investigated. It is shown that the optimal control is continuous in time with the values inL 2(Ω) and its time derivative belongs toL 2[OT×Ω].
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
J. Aubin, Approximation of Elliptic Boundary Value Problems, Wiley-Interscience, New York, 1972.
A. V. Balakrishnan, Applied Functional Analysis, Springer-Verlag, 1976.
V. Barbu, Constrained control problems with convex cost in Hilbert space,J. of Math Analysis 56, 529–547, 1976.
W. Borsage and McKnight, The Ritz Galerkin procedure for parabolic control problems,SIAM J. on Control, 11, 510–524, 1973.
N. Dunford and Y. Schwartz, Linear Operators,Interscience, 1, New York, 1958.
A. Friedman,Partial Differential Equations, Holt, Rinehart and Winston, Inc., New York, 1965.
D. Fujiwara, Concrete Characterization of the Domains of Fractional Powers of some Elliptic Differential Operators of the Second Order,Proc. Japan Acad., 43, 82–86, 1967.
I. Gelfand and S. Fomin, Calculus of Variations, Prentice-Hall, Inc., Englewood Cliffs, New Jersey, 1965.
W. Hager, Lipschitz continuity for constrained processes,SIAM J. on Control and Optimization, to appear.
W. Hager, The Ritz-Trefftz method for state and control constrained optimal control problems,SIAM J. on Numerical Analysis, 12, 854–867, 1975.
W. Hager and S. Mitter, Lagrange duality for the convex control problems,SIAM J. on Control, 14, 843–456, 1976.
E. Hille and R. Phillips: Functional Analysis and Semigroups, American Mathematical Society, Providence, Rhode Island, 1957.
S. Kurcyusz, On the existence and nonexistence of Lagrange's multipliers in Banach spaces.IOTA 20, 1, 1976.
I. Lasiecka and K. Malanowski, On regularity of solutions to convex optimal control problems with control constraints for parabolic systems,Control and Cybernetics, 6, 57–74, 1977.
J. Lions, Control Optimal des Systemes Gouvernes par des Equations aux Derivees Partielles. Dunod, Paris, 1968.
J. Lions and E. Magenes,Problemes aux Limites Nonhomogenes, 2. Dunod, Paris, 1968.
W. Rudin,Principles of Mathematical Analysis, McGraw-Hill Book Company, New York, 1964.
R. Sikorski, Advanced Calculus Functions of Several Variables, PWN, Warsaw, 1969.
J. Sokolowski, On parametric optimal control for weak solutions of abstract linear parabolic equations. Ph.D. Dissertation, Institute of Organization and Management, Warsaw, 1975.
L. De Simon, Un'applicazione della teoria degli integrali sigolari allo studio delle equazioni differenziali lineari astratte del primo ordine.Rendiconti del Seminario Mathematico della Universite di Padova, 34, 205–223, 1964.
D. Washburn, A semigroup approach to time optimal boundary control of diffusion processes, Ph.D. Dissertation, UCLA, 1974.
Author information
Authors and Affiliations
Additional information
Communicated by A. V. Balakrishnan
Research partially supported by National Aeronautics and Space Administration under Grant No. NSG 4015.
Rights and permissions
About this article
Cite this article
Lasiecka, I. State constrained control problems for parabolic systems: Regularity of optimal solutions. Appl Math Optim 6, 1–29 (1980). https://doi.org/10.1007/BF01442881
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01442881