Convexity properties of the minimum time function. (English) Zbl 0836.49013

Cannarsa, Piermarco; Sinestrari, Carlo

Calc. Var. Partial Differ. Equ. 3, No. 3, 273-298 (1995).

The paper deals with the minimum time optimal control problem governed by the system \[ \begin{aligned} y'(t) & = f \bigl( y(t), u(t) \bigr) \tag{*} \\ y(0) & = x \in \mathbb{R}^n, \quad u : [0, + \infty) \to U, \end{aligned} \] where \(U \subset \mathbb{R}^m\) is a compact set. Given a target \(K\) and denoting by \(\Omega \subset \mathbb{R}^n\) the set of points which can be driven in a finite time on \(K\), the minimum time function \(\tau : \Omega \to [0, + \infty)\) is defined as \[ \tau (x) : = \inf \biggl\{ T : \bigl( y(t), u(t) \bigr) \text{ solution of } (*),\;y(T) \in K \biggr\}. \] Under a Petrov type controllability assumption, it is proved that the semiconcavity of the distance function from the target (which can be considered as a mild regularity assumption on \(K)\) implies the semiconcavity of \(\tau\). For linear control systems a semiconvexity property can also be established.
These conditions can be used to study the structure (Hausdorff dimension estimates, propagation of singularities) of the set of nondifferentiability points of \(\tau\).

Reviewer: L.Ambrosio (Pavia)

MSC:

49L20	Dynamic programming in optimal control and differential games
49L25	Viscosity solutions to Hamilton-Jacobi equations in optimal control and differential games
93C10	Nonlinear systems in control theory

Keywords:

minimum time optimal control problem; Petrov type controllability; semiconcavity; semiconvexity; set of nondifferentiability points

References:

[1]	Alberti G., Ambrosio L., Cannarsa P.: On the singularities of convex functions. Manuscr. Math.76, 421–435 (1992) · Zbl 0784.49011 · doi:10.1007/BF02567770
[2]	Ambrosio L., Cannarsa P., Soner H.M.: On the propagation of singularities of semi-convex functions. Ann. Sc. Norm. Sup. Pisa.20, 597–616 (1993) · Zbl 0874.49041
[3]	Bardi M.: A boundary value problem for the minimum time function. SIAM J. Control Optim.27, 776–785 (1989) · Zbl 0682.49034 · doi:10.1137/0327041
[4]	Bardi M., Falcone M., An approximation scheme for the minimum time function. SIAM J. Control Optim.28, 950–965 (1990) · Zbl 0723.49024 · doi:10.1137/0328053
[5]	Bardi M., Soravia P.: Time-optimal control, Lie brackets and Hamilton-Jacobi equations. Preprint Univ. Padova, Dep. Math.13 (1991) · Zbl 0732.35013
[6]	Bressan A.: On two conjectures by Hájek. Funkcial. Ekvac.23, 221–227 (1980) · Zbl 0447.49035
[7]	Cannarsa P., Frankowska H.: Some characterizations of optimal trajectories in control theory. SIAM J. Control Optim.29, 1322–1347 (1991) · Zbl 0744.49011 · doi:10.1137/0329068
[8]	Cannarsa P., Soner H.M.: On the singularities of the viscosity solutions to Hamilton-Jacobi-Bellman equations. Indiana Univ. Math. J.36, 501–524 (1987) · Zbl 0612.70016 · doi:10.1512/iumj.1987.36.36028
[9]	Conti R.: Processi di controllo lineari in \(\mathbb{R}\) n (Quad. Unione Mat. Italiana 30) Bologna: Pitagora, 1985
[10]	Crandall M.G., Evans L.C., Lions P.-L.: Some properties of viscosity solutions of Hamilton-Jacobi equations. Trans. Amer. Math. Soc.282, 487–502 (1984) · Zbl 0543.35011 · doi:10.1090/S0002-9947-1984-0732102-X
[11]	Crandall M.G., Lions P.-L.: Viscosity solutions of Hamilton-Jacobi equations. Trans. Amer. Math. Soc.277, 1–42 (1983) · Zbl 0599.35024 · doi:10.1090/S0002-9947-1983-0690039-8
[12]	Evans L.C., James M.R.: The Hamilton-Jacobi-Bellman equation for time optimal control. SIAM J. Control Optim.27, 1477–1489 (1989) · Zbl 0688.49029 · doi:10.1137/0327076
[13]	Hájek O.: On differentiability of the minimal time function. Funkcial. Ekvac.20, 97–114 (1977)
[14]	Hermes H.: Feedback synthesis and positive local solutions to Hamilton-Jacobi-Bellman equations, in Proc. MTNS 87 (Sacks, Byrns, Martin editors). North Holland · Zbl 0673.49010
[15]	Hermes H., LaSalle J.P.: Functional analysis and time optimal control. New York: Academic Press 1969 · Zbl 0203.47504
[16]	Kruzhkov S.N.: Generalized solutions of the Hamilton-Jacobi equations of the eikonal type I. Math. USSR Sb.27, 406–445 (1975) · Zbl 0369.35012 · doi:10.1070/SM1975v027n03ABEH002522
[17]	Lions P.L.: Generalized solutions of Hamilton-Jacobi equations. Boston: Pitman 1982 · Zbl 0497.35001
[18]	Lee E.B., Markus L.: Foundations of optimal control theory. New York: John Wiley 1968
[19]	Petrov N.N.: Controllability of autonomous systems. Differential Equations4, 311–317 (1968) · Zbl 0234.93004
[20]	Petrov N.N.: On the Bellman function for the time-optimal process problem. J. Appl. Math. Mech34, 785–791 (1970) · Zbl 0253.49012 · doi:10.1016/0021-8928(70)90060-2
[21]	Sinestrari C: Semiconcavity of solutions of stationary Hamilton-Jacobi equations. Nonlinear Anal. (to appear) · Zbl 0836.49014
[22]	Sussmann H.J.: The structure of time-optimal trajectories for single-input systems in the plane: theC nonsingular case. SIAM J. Control Optim.25, 433–465 (1987) · doi:10.1137/0325025
[23]	Sussmann H.J.: The structure of time-optimal trajectories for single-input systems in the plane: the general real analytic case. SIAM J. Control Optim.25, 868–904 (1987) · Zbl 0664.93034 · doi:10.1137/0325048
[24]	Sussmann H.J.: Regular synthesis for time-optimal trajectories of single-input real analytic systems in the plane. SIAM J. Control Optim.25, 1145–1162 (1987) · Zbl 0701.93035 · doi:10.1137/0325062
[25]	Veliov V.M.: On the Lipschitz continuity of the value function in optimal control. J. Optim. Theory Appl. (to appear) · Zbl 0901.49022