Hostname: page-component-78c5997874-8bhkd Total loading time: 0 Render date: 2024-11-10T21:02:25.575Z Has data issue: false hasContentIssue false
  • English
  • Français

Nonconvex Duality and Semicontinuous Proximal Solutions of HJB Equationin Optimal Control

Published online by Cambridge University Press:  28 April 2009

Mustapha Serhani  and
Nadia Raïssi
Mustapha Serhani
Affiliation:
Département d'économie, FSJES, Université My Ismail, B.P. 3102, Toulal, Meknès, Maroc; mserhani@hotmail.com
Nadia Raïssi
Affiliation:
EIMA, Département de Mathématiques et d'Informatique, Faculté des Sciences, Université Ibn Tofail, B.P. 133, Kénitra, Maroc; n.raissi@lycos.com
Get access

Abstract

In this work, we study an optimal control problem dealing with differential inclusion.Without requiring Lipschitz condition of the set valued map, it isvery hard to look for a solution of the control problem. Our aim isto find estimations of the minimal value, (α), of the costfunction of the control problem. For this, we construct anintermediary dual problem leading to a weak duality result, andthen, thanks to additional assumptions of monotonicity of proximalsubdifferential, we give a more precise estimation of (α). Onthe other hand, when the set valued map fulfills the Lipshitzcondition, we prove that the lower semicontinuous (l.s.c.) proximalsupersolutions of the Hamilton-Jacobi-Bellman (HJB) equationcombined with the estimation of (α), lead to a sufficientcondition of optimality for a suspected trajectory. Furthermore, weestablish a strong duality between this optimal control problem anda dual problem involving upper hull of l.s.c. proximalsupersolutions of the HJB equation (respectively with contingentsupersolutions). Finally this strong duality gives rise to necessaryand sufficient conditions of optimality.

Keywords

Optimal controldualityHJB equationproximal supersolutionproximal subdifferential.
Type
Research Article
Information
RAIRO - Operations Research , Volume 43 , Issue 2 , April 2009 , pp. 201 - 214
Copyright
© EDP Sciences, ROADEF, SMAI, 2009

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

J.P. Aubin, Viability theory. Birkhäuser Boston Inc. (1991).
J.P. Aubin and A. Cellina, Differential inclusions. A Series of Comprehensive Studies in Mathematics, Springer-Verlag (1984).
J.P. Aubin and I. Ekeland, Applied nonlinear analysis. A Wiley Interscience Publication, John Wiley & Sons (1984).
J.P. Aubin and H. Frankowska, Set valued analysis. Birkhäuser (1990).
M. Bardi and I. Capuzzo-Dolcetta, Optimal control and viscosity solutions of Hamilton-Jacobi-Bellman equations. Birkhäuser (1997).
G. Barles, Solutions de viscosité des équations de Hamilton-Jacobi. Mathématiques & Applications, Springer-Verlag (1994).
E.N. Barron, Viscosity solutions and analysis in L∞, in Nonlinear analysis, differential equations and control, edited by F.H. Clarke and R. Stern. NATO Sciences Series (1999).
Barron, E.N., Evans, L.C. and Jensen, R., Viscosity solutions of Isaacs' equations and differential games with Lipshitz controls. J. Differ. Equ. 53 (1984) 213–233. CrossRef
Barron, E.N. and Jensen, R., Semicontinuous viscosity solutions for Hamilton-Jacobi equations with convex Hamiltonians. Commun. Partial Differ. Equ. 15 (1990) 1713–1742. CrossRef
Barron, E.N. and Jensen, R., Optimal control and semicontinuous viscosity solutions. Proc. Am. Math. Soc. 113 (1991) 397–402. CrossRef
C.W. Clark and J.M. Conrad, Natural resource economics. Cambridge University PressBirkhäuser (1995).
F.H. Clarke, Optimization and nonsmooth analysis. Society for Industrial and Applied Mathematics, Philadelphia (1983).
F.H. Clarke, Yu.S. Ledyaev and Stern R.J., Invariance, monotonicity, and applications. Nonlinear analysis, differential equations and control, edited by F.H. Clarke and R. Stern. NATO Sciences Series (1999).
Clarke, F.H., Ledyaev, Yu.S. and Wolenski, P.R., Qualitative properties of trajectories of control systems: a survey. J. Dyn. Control Syst. 1 (1995) 1–48. CrossRef
F.H. Clarke, Yu.S. Ledyaev, R.J. Stern and P.R. Wolenski, Nonsmooth analysis and control theory. Graduate Texts In Mathematics, Springer (1998).
Clarke, F.H. and Ledyaev, Yu.S., Mean value inequalities in Hilbert space. Trans. Am. Math. Soc. 344 (1994) 307–324. CrossRef
Clarke, F.H., Ledyaev, Yu.S. and Wolenski, P.R., Proximal analysis and minimization principles. J. Math. Anal. Appl. 196 (1995) 722–735. CrossRef
M.G. Crandall and P.L. Lions, Viscosity solutions of Hamilton-Jacobi equations. Trans. Am. Math. Soc. 277 (1983).
M.G. Crandall, L.C. Evan and P.L. Lions, Some properties of viscosity solutions of Hamilton-Jacobi equations. Trans. Am. Math. Soc. 282 (1984).
Evan, L.C. and James, M.R., The Hamilton-Jacobi-Bellman equation for time-optimal control. SIAM J. Control Optim. 27 (1989) 1477–1489. CrossRef
H. Frankowska, Equations de Hamilton-Jacobi contingentes. C. R. Acad. Sci. Série I 304 (1987).
Frankowska, H., Lower semicontinuous solutions of Hamilton-Jacobi-Bellman equations. SIAM J. Control Optim. 31 (1993) 257–272. CrossRef
Frankowska, H., Plaskacz, S. and Rzeżuchowski, T., Measurable viability theorem and the Hamilton-Jacobi-Bellman equations. J. Diff. Equ. 116 (1995) 265–305. CrossRef
Lions, P.L. and Souganidis, P.E., Differential Games, Optimal control and directional derivatives of viscosity solutions of Bellman's and Isaacs' equations. SIAM J. Control Optim. 23 (1985) 566–583. CrossRef
Penot, J.P. and Volle, M., Explicit solutions to Hamilton-Jacobi equations under mild continuity and convexity assumptions. J. Nonlinear Convex Anal. 1 (2000) 177–199.
N. Raïssi, Analyse proximale en optimisation. Ph.D. thesis, Université Montréal, Faculté des Arts et des Sciences, Canada (1987).
Rockafellar, T.R., Existence theorems for general control problem of Bolza and Lagrange. Adv. Math. 15 (1975) 312–337. CrossRef
M. Serhani and N. Raïssi, Nonconvex duality and generalized solutions of HJB equation in optimal control problem, in Conférence Internationale en Optimisation FGI, 4–9 Septembre, Montpellier, France (2000).
Serhani, M. and Raïssi, N., Nonconvex duality and viscosity solutions of HJB equation in optimal control problem. J. Convex Anal. 9 (2002) 625–648.
M. Serhani, Dualité et analyse non lisse : Optimisation non convexe, contrôle optimal et solutions généralisées de l'équation de HJB. Ph.D. thesis. University Ibn Tofail, Morocco (2002).
Vinter, R., Convex duality and nonlinear optimal control. SIAM J. Control Optim. 31 (1993) 518–538. CrossRef
Vinter, R. and Wolenski, P.R., Hamilton-Jacobi theory for optimal control problems with data measurable in time. SIAM J. Control Optim. 28 (1990) 1404–1419. CrossRef
Discontinuous, J.J. Ye solutions of the Hamilton-Jacobi equations for exit time problems. SIAM J. Control Optim. 38 (2000) 1067–1085.
L.C. Young, Lectures on the calculus of variations and optimal control theory. W.B. Saunders Company (1969).