if x(-) is an re-dimensional absolutely continuous vector ... 2 A is a constant set. .... A boat moves in a stream of constant velocity of magnitude s. The velocity of the boat relative to the stream is bounded ... The equations of motion of the boat are.
R. AGGARWAL Research Assistant.
G. LEiTMANN Professor of Engineering Science.
Afoidance Control1 An avoidance control is an admissible control 'which permits avoidance of a givenset of state space R". Existence theorems for avoidance controls are given for control processes governed by ordinary differential equations.
University of California, Berkeley, Calif.
Introduction
Ic
OST of the existing literature in the theory of control processes deals with the problem of finding a control which transfers a system's initial state to some terminal state belonging to a prescribed terminal manifold while minimizing a given performance index. However, sometimes we wish to prevent a dynamical system from attaining certain states. Thus an avoidance control is a control which permits the avoidance of a given set of states. In this paper we give existence theorems for avoidance controls for control processes governed by a set of ordinary differential equations.
Problem Statement We consider a control process described by a differential system in R" x(t) = f(x(t), u{t), I), x(t0) = x„
(1)
where [to, ti] is a compact interval in R and 1 / 0 (10) Therefore, Sx* = x*(t* - edt*) - x*(t*) = - tf(x*(i*),
u*(t*))St*
+ o(e) (11)
From (7), (9), and (11), for sufficiently small e > 0 \(i*)-5x*
= - e\(t*)-f(x*(t*),
u*(t*))5l* + o(e) = e5t* + o(e) > 0
(12)
c
Proof. Under hypotheses (a), (6), and (c) the set of reachable states K(t) is compact and convex and varies continuously with time on [to, li][l, p. 242]. Now let us suppose that there does not exist an admissible control u(-) which permits the avoidance of the set A during the interval [to, h]. Thus the trajectories corresponding to all admissible controls enter the set A during the interval (to, h]. Since A is a closed set with compact boundary dA and from our assumption all trajectories enter the set A during the interval (to, h,} by a modification of the proof of a theorem of Lee and Markus [1, p. 259] it is easy to prove t h a t there exists a time t* < h corresponding to an admissible control «*(•) with trajectory x*(-) such that x*(t*)G.dA and t* is the maximum of the first time to reach the set A. Also the endpoint x*(t*) of the trajectory x*(-) corresponding to timeoptimal control «*(•) belongs to the boundary of K(t*) [2, p. 182]. Now let X(-) = (Xo(-), MO) be a n (n + l)-component vector function which is a nontrivial solution of adjoint equations
Now, vector 5x* points into A ; therefore, by (12), X(i*) points into Ac. Since \(t*) points out of K{t*), set, K(i*) C .4. Case II: X0 = 0. Consider a trajectory £(•) generated by an admissible control u{t) = « £ f i , le[l* — ebt*, I*], St*, e > 0, and emanating from x*(t* — eSt*). Since t* is the maximum of the first time to enter A, x(i')GA for some t'G[t* — tSt*, I*] where I' = t* - e(St* - Si'), 0 < Si' < St*. Now, x(t') = x*(t*) - ef(x*(l*), u*(t*))8t* + ef(x*(t*), u(l*))St' + o(e) (13) Therefore, Sx = x{t') - x*(t*) = - tf{x*{t*),
+ ef(x*(t*), u(t*))St' + o(e)
(4)
, df(x*(l), u*(t)) w X(0 ' ^ ox
(5)
(14)
From (8) and X0 = 0 we obtain, \(t*)-f(x*(t*),
Xo(0 = 0
u*(t*))St*
u*(t*))
= 0
(15)
and then, in view of assumption (6), . X(0 = -
\(t*)-f(x*(t*),
u) < 0 for some uGSl
(16)
Then by (15) and (16), for sufficiently small e > 0,
where X0(0 = constant < 0. Let H(x, X, u) = - K + \;f(x, u)
(7)
Then by Pontryagin's Maximum Principle Max H{x*{t), \(t), u) = H(x*(t),
X(0, «-*(0)
«en
\(i*)-8x
(6)
(8)
and
Corollary 1. H(x*{t),
X(0, «*(0) = 0
(9)
i.e. on [to, ti\. If \ ( 0 = 0 for some T(k,
T)dT
(19)
) = (0, 1). At any time I the set of reachable states K(l) is a disc with center at (xfi + si, xfi) and radius (. Now we have two possibilities:
154 / J U N E 19 7 2
1
0 < s < 1
(23)
2
s > 1
(24)
First of all let us consider the case for which the stream speed 0 < s < 1. In this case (ui(t), u-2(t)) = (— s, 0) is admissible, so that (xi(t), x2(t)) = (0, 1). Hence there exists a velocity p r o . gram which permits avoidance of (a) the given bank parallel to stream, (6) the water fall, for all t G [(, •»). Now consider the case for which s > 1, say s = l i / 2 . (a) Let the bank A which we wish to avoid be given by^'x. > 1.4. Now K(t) — A ^ 4> for all t G [0, ]; hence, by Theorem 3 there exists a velocity program which permits avoidance of the given bank A. (b) Let the waterfall be given by the equation 2(x2)* + x, - 3 > 0
(25)
We find that K(t) - A ^
