Hostname: page-component-586b7cd67f-rdxmf Total loading time: 0 Render date: 2024-11-24T06:25:24.262Z Has data issue: false hasContentIssue false
  • English
  • Français

Global optimality conditions for a dynamic blocking problem

Published online by Cambridge University Press:  23 December 2010

Alberto Bressan  and
Tao Wang
Alberto Bressan
Affiliation:
Department of Mathematics, Penn State University, University Park, Pa. 16802, USA. [email protected]; [email protected]
Tao Wang
Affiliation:
Department of Mathematics, Penn State University, University Park, Pa. 16802, USA. [email protected]; [email protected]
Get access

Abstract

The paper is concerned with a class of optimal blocking problems in the plane. We consider a time dependent set R(t) ⊂ ℝ2, described as the reachable set for a differential inclusion. To restrict its growth, a barrier Γ can be constructed, in real time. This is a one-dimensional rectifiable set which blocks the trajectories of the differential inclusion. In this paper we introduce a definition of “regular strategy”, based on a careful classification of blocking arcs. Moreover, we derive local and global necessary conditions for an optimal strategy, which minimizes the total value of the burned region plus the cost of constructing the barrier. We show that a Lagrange multiplier, corresponding to the constraint on the construction speed, can be interpreted as the “instantaneous value of time”. This value, which we compute by two separate formulas, remains constant when free arcs are constructed and is monotone decreasing otherwise.

Keywords

Dynamic blocking problemoptimality conditionsdifferential inclusion with obstacles
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 18 , Issue 1 , January 2012 , pp. 124 - 156
Copyright
© EDP Sciences, SMAI, 2010

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 and A. Cellina, Differential inclusions. Springer-Verlag, Berlin (1984).
M. Bardi and I. Capuzzo-Dolcetta, Optimal control and viscosity solutions of Hamilton-Jacobi-Bellman equations. Birkhäuser, Boston (1997).
Boltyanskii, V.G., Sufficient conditions for optimality and the justification of the dynamic programming principle. SIAM J. Control Optim. 4 (1966) 326–361. Google Scholar
Bressan, A., Differential inclusions and the control of forest fires. J. Differ. Equ. 243 (2007) 179–207. Google Scholar
Bressan, A. and De Lellis, C., Existence of optimal strategies for a fire confinement problem. Comm. Pure Appl. Math. 62 (2009) 789–830. Google Scholar
Bressan, A. and Hong, Y., Optimal control problems on stratified domains. NHM 2 (2007) 313–331. Google Scholar
A. Bressan and B. Piccoli, Introduction to the Mathematical Theory of Control, AIMS Series in Applied Mathematics 2. AIMS, Springfield Mo. (2007).
Bressan, A. and Wang, T., The minimum speed for a blocking problem on the half plane. J. Math. Anal. Appl. 356 (2009) 133–144. Google Scholar
Bressan, A. and Wang, T., Equivalent formulation and numerical analysis of a fire confinement problem. ESAIM : COCV 16 (2010) 974–1001. Google Scholar
Bressan, A., Burago, M., Friend, A. and Jou, J., Blocking strategies for a fire control problem. Analysis and Applications 6 (2008) 229–246. Google Scholar
Brunovský, P., Every normal linear system has a regular time-optimal synthesis. Math. Slovaca 28 (1978) 81–100. Google Scholar
F.H. Clarke, Optimization and Nonsmooth Analysis. Second edition, SIAM, Philadelphia (1990).
W.H. Fleming and R.W. Rishel, Deterministic and Stochastic Optimal Control. Springer-Verlag, New York (1975).
A.D. Ioffe and V.M. Tihomirov, Theory of Extremal Problems. North-Holland, New York (1974) 233–235.
R. Vinter, Optimal Control. Birkhäuser, Boston (2000).