no. 2
Nonconvex duality and semicontinuous proximal solutions of HJB equation in optimal control
RAIRO - Operations Research - Recherche Opérationnelle, Tome 43 (2009) no. 2, pp. 201-214.

In this work, we study an optimal control problem dealing with differential inclusion. Without requiring Lipschitz condition of the set valued map, it is very hard to look for a solution of the control problem. Our aim is to find estimations of the minimal value, (α), of the cost function of the control problem. For this, we construct an intermediary dual problem leading to a weak duality result, and then, thanks to additional assumptions of monotonicity of proximal subdifferential, we give a more precise estimation of (α). On the other hand, when the set valued map fulfills the Lipshitz condition, we prove that the lower semicontinuous (l.s.c.) proximal supersolutions of the Hamilton-Jacobi-Bellman (HJB) equation combined with the estimation of (α), lead to a sufficient condition of optimality for a suspected trajectory. Furthermore, we establish a strong duality between this optimal control problem and a dual problem involving upper hull of l.s.c. proximal supersolutions of the HJB equation (respectively with contingent supersolutions). Finally this strong duality gives rise to necessary and sufficient conditions of optimality.

DOI : 10.1051/ro/2009012
Classification : 49J24, 49K24, 49N15, 49L99, 49J52
Mots clés : optimal control, duality, HJB equation, proximal supersolution, proximal subdifferential 
@article{RO_2009__43_2_201_0,
     author = {Serhani, Mustapha and Ra{\"\i}ssi, Nadia},
     title = {Nonconvex duality and semicontinuous proximal solutions of {HJB} equation in optimal control},
     journal = {RAIRO - Operations Research - Recherche Op\'erationnelle},
     pages = {201--214},
     publisher = {EDP-Sciences},
     volume = {43},
     number = {2},
     year = {2009},
     doi = {10.1051/ro/2009012},
     mrnumber = {2527863},
     zbl = {1162.49004},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/ro/2009012/}
}
TY  - JOUR
AU  - Serhani, Mustapha
AU  - Raïssi, Nadia
TI  - Nonconvex duality and semicontinuous proximal solutions of HJB equation in optimal control
JO  - RAIRO - Operations Research - Recherche Opérationnelle
PY  - 2009
SP  - 201
EP  - 214
VL  - 43
IS  - 2
PB  - EDP-Sciences
UR  - http://www.numdam.org/articles/10.1051/ro/2009012/
DO  - 10.1051/ro/2009012
LA  - en
ID  - RO_2009__43_2_201_0
ER  - 
%0 Journal Article
%A Serhani, Mustapha
%A Raïssi, Nadia
%T Nonconvex duality and semicontinuous proximal solutions of HJB equation in optimal control
%J RAIRO - Operations Research - Recherche Opérationnelle
%D 2009
%P 201-214
%V 43
%N 2
%I EDP-Sciences
%U http://www.numdam.org/articles/10.1051/ro/2009012/
%R 10.1051/ro/2009012
%G en
%F RO_2009__43_2_201_0
Serhani, Mustapha; Raïssi, Nadia. Nonconvex duality and semicontinuous proximal solutions of HJB equation in optimal control. RAIRO - Operations Research - Recherche Opérationnelle, Tome 43 (2009) no. 2, pp. 201-214. doi : 10.1051/ro/2009012. http://www.numdam.org/articles/10.1051/ro/2009012/

[1] J.P. Aubin, Viability theory. Birkhäuser Boston Inc. (1991). | MR | Zbl

[2] J.P. Aubin and A. Cellina, Differential inclusions. A Series of Comprehensive Studies in Mathematics, Springer-Verlag (1984). | MR | Zbl

[3] J.P. Aubin and I. Ekeland, Applied nonlinear analysis. A Wiley Interscience Publication, John Wiley & Sons (1984). | MR | Zbl

[4] J.P. Aubin and H. Frankowska, Set valued analysis. Birkhäuser (1990). | MR | Zbl

[5] M. Bardi and I. Capuzzo-Dolcetta, Optimal control and viscosity solutions of Hamilton-Jacobi-Bellman equations. Birkhäuser (1997). | MR | Zbl

[6] G. Barles, Solutions de viscosité des équations de Hamilton-Jacobi. Mathématiques & Applications, Springer-Verlag (1994). | MR | Zbl

[7] 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). | MR | Zbl

[8] E.N. Barron, L.C. Evans and R. Jensen, Viscosity solutions of Isaacs' equations and differential games with Lipshitz controls. J. Differ. Equ. 53 (1984) 213-233. | MR | Zbl

[9] E.N. Barron and R. Jensen, Semicontinuous viscosity solutions for Hamilton-Jacobi equations with convex Hamiltonians. Commun. Partial Differ. Equ. 15 (1990) 1713-1742. | MR | Zbl

[10] E.N. Barron and R. Jensen, Optimal control and semicontinuous viscosity solutions. Proc. Am. Math. Soc. 113 (1991) 397-402. | MR | Zbl

[11] C.W. Clark and J.M. Conrad, Natural resource economics. Cambridge University PressBirkhäuser (1995).

[12] F.H. Clarke, Optimization and nonsmooth analysis. Society for Industrial and Applied Mathematics, Philadelphia (1983). | MR | Zbl

[13] 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). | MR | Zbl

[14] F.H. Clarke, Yu.S. Ledyaev and P.R. Wolenski, Qualitative properties of trajectories of control systems: a survey. J. Dyn. Control Syst. 1 (1995) 1-48. | MR | Zbl

[15] F.H. Clarke, Yu.S. Ledyaev, R.J. Stern and P.R. Wolenski, Nonsmooth analysis and control theory. Graduate Texts In Mathematics, Springer (1998). | MR | Zbl

[16] F.H. Clarke and Yu.S. Ledyaev, Mean value inequalities in Hilbert space. Trans. Am. Math. Soc. 344 (1994) 307-324. | MR | Zbl

[17] F.H. Clarke, Yu.S. Ledyaev and P.R. Wolenski, Proximal analysis and minimization principles. J. Math. Anal. Appl. 196 (1995) 722-735. | MR | Zbl

[18] M.G. Crandall and P.L. Lions, Viscosity solutions of Hamilton-Jacobi equations. Trans. Am. Math. Soc. 277 (1983). | MR | Zbl

[19] 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). | MR | Zbl

[20] L.C. Evan and M.R. James, The Hamilton-Jacobi-Bellman equation for time-optimal control. SIAM J. Control Optim. 27 (1989) 1477-1489. | MR | Zbl

[21] H. Frankowska, Equations de Hamilton-Jacobi contingentes. C. R. Acad. Sci. Série I 304 (1987). | MR | Zbl

[22] H. Frankowska, Lower semicontinuous solutions of Hamilton-Jacobi-Bellman equations. SIAM J. Control Optim. 31 (1993) 257-272. | MR | Zbl

[23] H. Frankowska, S. Plaskacz and T. Rzeżuchowski, Measurable viability theorem and the Hamilton-Jacobi-Bellman equations. J. Diff. Equ. 116 (1995) 265-305. | MR | Zbl

[24] P.L. Lions and P.E. Souganidis, Differential Games, Optimal control and directional derivatives of viscosity solutions of Bellman's and Isaacs' equations. SIAM J. Control Optim. 23 (1985) 566-583. | MR | Zbl

[25] J.P. Penot and M. Volle, Explicit solutions to Hamilton-Jacobi equations under mild continuity and convexity assumptions. J. Nonlinear Convex Anal. 1 (2000) 177-199. | MR | Zbl

[26] N. Raïssi, Analyse proximale en optimisation. Ph.D. thesis, Université Montréal, Faculté des Arts et des Sciences, Canada (1987).

[27] T.R. Rockafellar, Existence theorems for general control problem of Bolza and Lagrange. Adv. Math. 15 (1975) 312-337. | MR | Zbl

[28] 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).

[29] M. Serhani and N. Raïssi, Nonconvex duality and viscosity solutions of HJB equation in optimal control problem. J. Convex Anal. 9 (2002) 625-648. | MR | Zbl

[30] 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).

[31] R. Vinter, Convex duality and nonlinear optimal control. SIAM J. Control Optim. 31 (1993) 518-538. | MR | Zbl

[32] R. Vinter and P.R. Wolenski, Hamilton-Jacobi theory for optimal control problems with data measurable in time. SIAM J. Control Optim. 28 (1990) 1404-1419. | MR | Zbl

[33] J.J. Ye, Discontinuous solutions of the Hamilton-Jacobi equations for exit time problems. SIAM J. Control Optim. 38 (2000) 1067-1085. | MR | Zbl

[34] L.C. Young, Lectures on the calculus of variations and optimal control theory. W.B. Saunders Company (1969). | MR | Zbl

Cité par Sources :