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Optimality Conditions and Synthesis for the Minimum Time Problem

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Journal of Mathematical Systems, Estimation, and Control c 1998 Birkhauser-Boston Vol. 8, No. 1, 1998, pp. 123{126

SUMMARY

Optimality Conditions and Synthesis

for the Minimum Time Problem



P. Cannarsa, y H. Frankowska y ; C. Sinestrari y

1 Introduction

In this paper we are concerned with the minimum time optimal control problem for the system

 y 0( t) =f(y(t);u(t)); t0; y(0) =x; (1.1) where x 2 IR n and f : IR n U ! IR n are given, U being a complete

separable metric space. We assume thatf is continuous and 8 > > < > > : i) 8R>0; 9c R >0 such that 8u2U; f(;u) is c R ,Lipschitz on B R(0) ii) 9k>0 such that 8x2IR n ; sup u2U jf(x;u)j  k(1 +jxj): (1.2) A measurable function u : [0;+1[! U is called a control and the

corresponding solution of the state equation (1:1) is denoted byy(;x;u).

Given a nonempty closed set K IR

n (called the target) and a point x2K

c, we are interested in minimizing, over all controls

u, the time taken

for the solutiony(;x;u) to reachK. The value function, denoted by T(x) = inf

u

ft0 :y(t;x;u)2Kg (1.3)

asxranges overR

n, is called the Minimum Time function of the problem. Received March 3, 1996; received in nal form November 4, 1996. Full electronic

manuscript (published January 1, 1998) = 24 pp, 507,956 bytes. Retrieval Code: 61346

yThis research was supported by the Human Capital and Mobility Programme of the

European Community through the network on Calculus of Variations and Control of Constrained Uncertain Systems.

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P. CANNARSA, H. FRANKOWSKA, C. SINESTRARI

It is well known that T may be discontinuous even for very smooth

data. However, if in addition

8x2IR n

; f(x;U) are closed and convex, (1.4)

thenT is lower semicontinuous.

In this paper we focus our attention on necessary and sucient optimal-ity conditions. In particular, we are interested in extending to time optimal control the results of the rst two authors ([2]) for the Mayer problem.

2 Optimality Conditions

For any non{empty setS IR

n, we denote by S

c its complement, by S

,

its (negative) polar byT S(

x) the contingent cone to S at a point x 2S,

and by S(

x) the set of perpendiculars toS atx.

Let': IR n

7!IR[f1gbe an extended function, and let x02IR n

be such that'(x0)6=1. We set

D+'(x0) =  p2IR n j limsup x!x 0 '(x),'(x0),hp;x,x0i jx,x0j  0  ; and, for any vectorv2IR

n, D # '(x0)(v) = limsup h!0+;v 0 !v '(x0+hv 0) ,'(x0) h :

Finally, for any Lipschitz arcz: [a;b]!IR

n and any t2[a;b[ we set Dz(t) := Limsup s#t z(s),z(t) s,t

where Limsup denotes the Painleve{Kuratowski upper limit [1].

We begin with a re ned version of Pontryagin's Maximum Principle. Assume (1:2) and suppose thatf is di erentiable with respect to x. Let



ube an optimal control for problem (1:3) at some point x0 and setT0 = T(x0); y(t) =y(t;x0;u). Then, for every p0 2T

K

c(y(T0))

, we show that

the solutionp: [0;T0]!IR

n of the adjoint system ,p 0( t) =  @f @x (y(t);u(t))  ? p(t); p(T0) = ,p0 (2.5)

satis es the minimum principle

hp(t);vi = min u2U

hp(t);f(y(t);u)i (2.6)

for allt2[0;T0] and all vectorsv2 Limsup s!t

 y(s),y(t)

s,t :

We also derive a co{state inclusion of the type obtained in [3] and [2] for the (Lipschitz) value function of Bolza and Mayer problems. We recall that the HamiltonianH associated to control system (1.1) is given by

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SUMMARY H(x;p) = sup u2U hp; f(x;u)i; 8x;p2IR n : (2.7)

Let (y0;u0) be an optimal pair at a point x0 2 K

c and set T0 = T(x0): Suppose that 2 K c( y0(T0)) is such that H(y0(T0);)>0: (2.8)

Then, we prove that the solutionpof problem (2:5) with p0=H(y0(T0);) ,1  satis es p(t)2D+T(y0(t)); 8t2[0;T0[: (2.9) Furthermore, since K c(y0(T0))T K c(y0(T0)) , ;we conclude that, if 2 K

c(y0(T0)) and (2.8) holds true, then the solution of the adjoint system

(2:5) withp0=H(y0(T0);) ,1

 satis es both the minimum principle (2.6)

and the co{state inclusion (2.9).

From the above necessary conditions we derive the following necessary and sucient optimality result.

Theorem 2.1

Assume (1:2);(1:4), and suppose that f is di erentiable

with respect tox. Letx02K c

;T0>0;and letu0() be a xed control such

that the corresponding trajectory y0() =y(;x0;u0) satis es y0(t)2=K for

all t2[0;T0[; y0(T0)2K ; and(2:8) for some  2 K

c(y0(T0)): Then, u0

is time optimal if and only if

D #

T(y0(t))(v) =,1; 8v2Dy0(t); 8t2[0;T0[:

3 Time Optimal Synthesis

Let us assume that f is di erentiable with respect to x and consider the

set{valued maps

U(x) = 

fu2U j h p;f(x;u)i=,1; 8p2D+T(x)g ifD+T(x)6=;

; otherwise

andF(x) =f(x;U(x)). It is not dicult to verify that if (1.4) holds true,

thenF has closed convex, possibly empty, images.

Suppose that y is an optimal trajectory at a point x 2 K c

; set T0 = T(x), and let H(y(T0);)>0 for some vector 2

K c(

y(T0)):Then, we

show that yis a solution of the di erential inclusion  y 0( t) 2 F(y(t)); a.e. t2[0;T0] y(0) = x (3.10) and minp2D + T(y(t))

jpjM; 8t2[0;T0[:for some constantM>0.

Conversely, we obtain the following result. 125

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P. CANNARSA, H. FRANKOWSKA, C. SINESTRARI

Theorem 3.1

Assume(1:2);(1:4) and let x 2 K c

;T0 > 0: Suppose that y() is a solution of (3:10) satisfyingy(t)2=Kfor allt2[0;T0[; y(T0)2K ;

min

p2D +

T(y(t))

jpj  M; 8t2[0;T0[ (3.11)

for some constant M>0. Then y is time optimal.

Moreover, we can drop assumption (3.11) if, instead of an absolutely con-tinuous solution of (3.10), we consider a contingent solution of it, i.e. a continuous arcy such that

Dy(t)\F(y(t))=6 ;; 8t2[0;T0[:

Finally, we consider another time optimal feedback in the form

G(x) =  f(x;u)ju2U; 9p2D + T(x) : hp;f(x;u)i=,1 ;

and the di erential inclusion

 y 0( t) 2 G(y(t)) y(0) = x: (3.12)

Theorem 3.2

Assume (1:2);(1:4), and suppose that f is di erentiable

with respect to x. Let x 2 K c

;T0 > 0; and let u() be a xed

con-trol. Suppose that trajectory y() = y(;x;u) satis es y(t) 2= K for all t 2[0;T0[; y(T0)2 K ; and (2:8) for some  2 

K c(

y(T0)): Then, u() is

time optimal if and only if y is a contingent solution of(3:12) in [0;T0]:

References

[1] J.-P. Aubin and H. Frankowska. Set-Valued Analysis. Basel: Birkhauser, 1990.

[2] P. Cannarsa and H. Frankowska. Some characterizations of optimal tra-jectories in control theory, SIAM J. on Control and Optimization,

29

(1991), 1322{1347.

[3] F.H. Clarke and R.B. Vinter. The relationship between the maximum principle and dynamic programming, SIAM J. Control Optim.,

25

(1987), 1291{1311.

Dipartimento di Matematica, Universita di Roma \Tor Ver-gata," Via della Ricerca Scientifica, 00133 Roma, Italy

CEREMADE, CNRS, Universite Paris-Dauphine 75775 Paris Cedex 16, France

Dipartimento di Matematica, Universita di Roma \Tor Ver-gata," Via della Ricerca Scientifica, 00133 Roma, Italy

Communicated by Anders Lindquist

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