Abstract
We consider optimal problems for a general nonlinear nonconvex input-output relation for Banach space valued functions. A maximum principle is obtained using Ekeland's variational principle. The formulation applies to systems described by ordinary differential equations, functional differential equations, and partial differential equations (both for distributed and boundary control systems).
Similar content being viewed by others
References
Alexander J PhD Dissertation, University of California, Los Angeles
Barbu V (1984) Optimal Control of Variational Inequalities. Research Notes in Mathematics, No 100. Pitman, London
Barbu V (1986) The time optimal problem for a class of nonlinear distributed systems. Proceedings of the IFIP Workshop on Control Problems for Systems Described for Partial Differential Equations, Gainesville
Clarke (1976) The maximum principle with minimum hypotheses. SIAM J Control Optim 14:1078–1091
Clarke F (1983) Optimization and Nonsmooth Analysis. Wiley, New York
Colonius F, Hinrichsen D (1978) Optimal control of functional differential systems. Siam J. Control Optim 16:861–879
Dieudonné J (1969) Foundations of Modern Analysis. Academic Press, New York
De Simon L (1964) Un' applicazione della teoria degli integrali singolari allo studio delle equazioni differenziali lineari astratte del primo ordine. Rend Sem Mat Univ Padova 34:205–223
Egorov JuV (1964) Necessary conditions for the optimality of control in Banach spaces. Mat Sb 64:79–101
Ekeland I (1972) Sur les problémes variationnels. CR Acad Sci Paris 275:1057–1059
Ekeland I (1979) Nonconvex minimization problems. Bull Amer Math Soc (NS) 1:443–474
Fattorini HO (1964) Time-optimal control of solutions of operational differential equations. SIAM J Control Optim 2:54–59
Fattorini HO (1966) Control in finite time of differential equations in Banach spaces. Comm Pure Appl Math 19:17–34
Fattorini HO (1967) On Jordan operators and rigidity of linear control systems. Rev Unión Mat Argentina 23:67–75
Fattorini HO (1974) The time optimal control problem in Banach spaces. Appl Math Optim 1:163–188
Fattorini HO (1975) Local controllability of a nonlinear wave equation. Math Systems Theory 9:30–44
Fattorini HO (1975) Exact Controllability of Linear Systems in Infinite Dimensional Spaces. Lecture Notes in Mathematics, vol 466. Springer-Verlag, Heidelberg, pp 166–183
Fattorini HO (1976) The time optimal control for boundary control of the heat equation. In: Russell DL (ed) Calculus of Variations and Control Theory. Academic Press, New York, pp 305–319
Fattorini HO (1977) The time optimal problem for distributed control of systems described by the wave equation. In: Aziz AK, Wingate JW, Balas MJ (eds) Control Theory of Systems Governed by Partial Differential Equations. Academic Press, New York, pp 151–175
Fattorini HO (1984) The Maximum Principle for Nonlinear Nonconvex Systems in Infinite Dimensional Spaces. Lectures Notes in Control and Information Sciences, vol 75 (1986). Springer-Verlag, New York, pp 162–178
Fattorini HO (1985) The maximum principle for nonlinear nonconvex systems with set targets. Proceedings of the 24th IEEE Conference on Decision and Control, Fort Lauderdale, pp 1999–2004
Fattorini HO (1986) Optimal control of nonlinear systems: convergence of suboptimal controls, I. Proceedings of the Special Session on Operator Methods in Optimal Control Problems (Annual AMS Meeting, New Orleans)
Fattorini HO (1986) Optimal control of nonlinear systems: convergence of suboptimal controls, II. Proceedings of the IFIP Workshop on Control Problems for Systems Described for Partial Differential Equations, Gainesville
Friedman A (1964) Partial Differential Equations of Parabolic Type. Prentice-Hall, Englewood Cliffs, NJ
Gabasov P, Kirillova F (1971) Qualitative Theory of Optimal Processes. Isdatelstvo “Nauka”, Moscow
Haraux A (1981) Nonlinear Evolutions Equations-Global Behavior of Solutions. Lecture Notes in Mathematics, vol 841. Springer-Verlag, Heidelberg
Henry D (1981) Geometric Theory of Semilinear Parabolic Equations. Lecture Notes in Mathematics, vol 840. Springer-Verlag, Heidelberg
Ioffe AI, Tikhomirov VM (1974) Theory of Extremal Problems. Isdatelstvo “Nauka”, Moscow
Li Xunjing, Yao Yunlong (1981) On optimal control of distributed parameter systems. Proceedings of the Third IFAC Triennial World Congress, Kyoto, pp 207–121
Li Xunjing, Yao Yunlong (1984) Maximum principle of distributed parameter systems with time lags. Proceedings of the Second International Conference of Control Theory of Distributed Parameter Systems, Vorau
Pontryagin LS, Boltyanskii VG, Gamkrelidze RV, Mischenko EF (1961) Mathematical Theory of Optimal Processes. Goztekhizdat, Moscow (English translation (1962) Wiley, New York)
Sachs E (1978) A parabolic control problem with a boundary condition of the Stefan-Boltzmann type. Z Angew Math Mech 58:443–449
Sachs E (1983) Optimal control problems in diffusion processes with a nonsmooth boundary condition. Math Methods Appl Sci 5:117–130
Weissler FB (1979) Semilinear evolution equations in Banach spaces. J Funct Anal 32:277–296
Wolfersdorf LV (1975) Optimale Steuerung einer Klasser nichtlineare Aufheizungsprozesse. Z Angew Math Mech 55:353–362
Wolfersdorf LV (1975) Optimale Steuerung bei Hammersteinschen Integralgleichungen mit schwach singulären Kernen. Math Operationsforsch Statist 6:609–625
Yao Yunlong (1982) Maximum principle of semi-linear distributed systems. Proceedings of the Third IFAC Symposium on Control of Distributed Parameter Systems, pp 81–86
Zowe J, Kurcyusz S (1979) Regularity and stability for the mathematical programming problem in Banach spaces. Appl Math Optim 5:49–62
Author information
Authors and Affiliations
Additional information
This work was supported in part by the National Science Foundation under Grant No. DMS-8200645.
Rights and permissions
About this article
Cite this article
Fattorini, H.O. A unified theory of necessary conditions for nonlinear nonconvex control systems. Appl Math Optim 15, 141–185 (1987). https://doi.org/10.1007/BF01442651
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/BF01442651