THE UNIVERS ITY OF MICHIGAN COLLEGE OF LITERATURE, SCIENCE, AND THE ARTS Department of Mathematics Technical Report No. 5 EXISTENCE THEOREMS FOR f LINEAR IN u OR IN x Lamberto Cesari ORA Project 02416 submitted for: UNITED STATES AIR FORCE AIR FORCE OFFICE OF SCIENTIFIC RESEARCH GRANT NO. AFOSR-69-1662 ARLINGTON, VIRGINIA administered through: OFFICE OF RESEARCH ADMINISTRATION ANN ARBOR August 1969

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5. EXISTENCE THEOREMS FOR f LINEAR IN u OR IN x 5o 1. LINEAR PROBLEMS We briefly consider in this chapter problems of minimum in which the system has some linear character, say f(t,x,u) = (fl,,f ) is either linear in In u, or in x, or both, dx/dt = B(t,x) u + C(t,x), (5.1.1) dx/dt = A(t,u) x + C(t,u), (5.1.2) dx/dt = A(t) x + B(t) u + C(t), (5.1.3) while the functional has one of the usual forms I[x,u] = g(tl,x(tl), t2,x(t2)), (5.1.4) I[x,u] = ft2 f (t,x(t),u(t))dt. (5.1.5) In case the functional has the Lagrange form (5.1.5), the particular cases f = B (t,x) u + C (t,x) (5.1.6) f = A (t,u) x + C (t,u) (5.1.7) f = A (t) x + B (t) u + C (t) (5.1.8) are of some interest. Above A,B,C denote matrices of the types n x n, n x m, n x 1 respectively, and A, B, C of the types 1 x n, 1 x m, 1 x 1. Problems as above are all particular cases of those considered in Chapter 3 and 4, and the existence theorems already proved apply. Nevertheless, for problems with f linear in u (systems (5.1.1) and 1

(5.1.3), functionals (5.1.4) or (5.1.5),we list below (5.2) a few corollaries. For problems with both f and f linear in x, namely of the forms f = A(t) x + C(t,u), f = A (t) x + C (t,u) (particular cases of (5.1.2), (5.1.3), (5.1.7), (5.1.8)), or alternatively for f as stated and functional (5.1.4), a completely different theorem will be stated and proved in (5.3). 5.2. EXISTENCE THEOREMS FOR f LINEAR IN u As usual, A denotes a subset of the tx-space E B a subset of the l+n' tlxlt2x2 - space E2n+2, U(t,x) a subset of the u-space E, g, f are scalars, f an n-vector, and M,Q,Q are the usual sets M = [(t,x,u)l(t,x) c A, u E U(t,x)], Q(t,x) = f(t,x,U(t,x)) c E and Q(t,x) = [z = (z,z)l = f (t,x,u), z f(t,x,u), u E U(t,x)] E +. Finally, 0 denotes a class of admissible pairs. (5.2.i) (A corollary of theorem 1). If A is compact, B closed, M compact, f = B(t,x) u + C(t,x), B,C matrices with entries continuous on A, g(tl,xl,t2,x2) continuous on B, Q closed and not empty, then the functional (5.1.4) has an absolute minimum in Q. If A is not compact but closed, then (5.2.i) still holds under conditions (a), (b), or (a), (b), (c), of (3.3), according as A is contained in a slab [(t,x)lt < t < T, x c E ], t, T finite, or not. If A is contained in such a slab, and the entries of the matrices B = (bij(t,x)), C = (ci(t,x)) satisfy relations lbi (t,x)l, Ici(t,x)l < H + KIxI for some constant H,K and J all (t,x) E A, then certainly condition (a ) of (3.3) is satisfied, and so is (a). If A is not contained in such a slab, it is enough to know that the relations above for all (t,x) E A with |tl < N and constants H,K depending on N. This is certainly the case if f is of the form (5.1.3) with matrices A(t), 2

B(t), C(t) continuous on the real axis. For alternate conditions see (3.3). (5.2.ii) (A corollary of Theorem 2). If A is compact, B closed, M compact, f = B(t,x) u + C(t,x), B,C matrices with entries continuous on A, f (t,x,u) continuous on M and convex in u, Q closed and not empty, then the functional (5.1.5) has an absolute minimum in Q. If A is not compact but closed, then (5.2.ii) still holds under conditions (a), (b), or (a'), (b), (c), of (3.4), and the same remarks hold as before. For alternate conditions see again (3.4). If f is linear in u, say of the form (5.1.6) or (5.1.8), then certainly f is convex in u. (5.2.iii) (A Corollary of Theorem 3). Let A be compact, B closed, M closed, f = B(t,x) u + C(t,x), B,C continuous matrices with entries continuous on A, f (t,x,u) continuous on M, convex in u, satisfying (W) f (txu) > (t), where 4 is a locally L-integrable given function. Let us assume that the sets Q(t,x) satisfy condition (Q) at all (t,x) e A with exception perhaps of a set of points in A whose t coordinate lie in a set of measure zero on the t-axis. Let Q1 be a closed nonempty family of admissible pairs satisfying a relation t2 I|x'(t)lP dt < L for some p > 1 and L > 0. Then the functional (5.1.5) has tj an absolute minimum in Q. Condition (r) can be replaced by the much weaker condition (r*) of (2.10). Condition (Q) and condition (4*) and (r) are certainly satisfied under conditions (a) and (X) of (2.12). (The last situation is an extension of the socalled "normal convexity" case for free problems.) If A is not compact, but A is closed, then theorem (5.2.iii) (and variants above) still holds provided condition (b), or conditions (b), (a') of (4.3) 15

hold, according as A is contained in a slab [t < t T, x E E ], t T finite, or not. For alternate conditions see (4.3). (5.2.iv) (A Corollary of Theorem 3). Let A be compact, B closed, M closed, f = B(t,x) u + C(t,x), B,C continuous matrices with entries continuous on A, f (t,x,u) continuous on M, and convex in u. Let us assume that the sets Q(t,x) satisfy condition (Q) at all (t,x) c A with exception perhaps of a set of points in A whose t coordinate lie in a set of measure zero on the t-axis. Let us assume that the following growth condition is satisfied: (y) given c > 0 there is a locally integrable function E~(t) such that [B(t,x) u + C(t,x)] < 4 (t) + ~ f (t,x,u) for all (t,x,u) E M. If Q is a closed nonempty family of admissible pairs, then the functional (5.1.5) has an absolute minimum in Q. Condition (Q) is certainly satisfied under conditions (a) and (X) of (2.12), this situation being an extension of the so called "normal convexity" case for free problems. If a growth condition hold as (0): there is a scalar continuous function C(O), O < < < + a, such that 0( )/5 + + as e +a, and f (t,x,u) >'(]u[) for all (t,x,u) E M, then certainly condition (y) holds, and all sets Q(t,x) have property (Q). If A is not compact, but A is closed, then theorem (5.2.iv) (and variants) still holds provided conditions (a), (b), or conditions (a'), ( b ), (c') of (4.3) hold, according as A is contained in a slab [t < t < T, x c E ], t, T finite, or not. For alternate conditions, in particular conditions (aa'), see (4-3). Analogous corollaires of theorem 4 and others are left as exercises for the reader. 4

5.3. EXISTENCE THEOREMS FOR f LINEAR IN x Let us consider first problems with system of the form (5.1.2), or dx/dt = A(t,u) x + C(t,u) Existence theorems 1 to 4 of Chapters 1 and 2 naturally apply. However, unlike the case of f linear in u, these theorems do not yield statements which are in any way simpler than the original ones, and therefore, we do not restate them. Of some interest may be the remark that, if the entries of the matrices A = (a. (t,u)) and C = (ci(t,u)) are bounded, say laijl < M, Icil < M, than the condition (a) of (4.3) is satisfied: n ij i' xfl +...+ x f = aij x x + E CiX < M E Ix x2 + M 0. lx i n Jij i i — 0 ij 0 1 <M {(Ixl +... + M i ((xi) +1) < (M n + M )(.(x) + 1). An analogous remark holds for condition (a'). Let us consider now problems where the system has the more particular form dx/dt = A(t) x + C(t,u) (5.3.1) with compact control space U, A closed of the form A = [t,T] x En, or A = E1 x En, and functional of the (Mayer) form (5.1.4). (Below we shall consider the analogous case with functional of the form (5.1.5) and f = A (t) x + C (t,u)). 5

(5.3.i) Existence Theorem (for Mayer problems and linear systems (5.3.1)). Let A = [t,T] x En, to, T finite, U a fixed compact set of the u-space E, f = A(t) x + C(t,u), A,C matrices with entries continuous in [t,T] and [t,T] x U respectively. Let B be a closed subset of the tlxlt2x2-space and g(tl,xl,t2,x2) a scalar function continuous on B. Let P be a compact subset of A, and let r be the class, which we suppose not empty, of all admissible pairs x,u whose trajectory x possesses at least one point (t*, x(t*)) E P. Then the functional (5.1.4) has an absolute minimum in Q. If A = E1 x E then (5.3.i) still holds under the additional condition (d) of (3.3) (or alternates). Note that conditions (a) and (a') hold here automatically as consequences of the hypotheses (U compact, A,C continuous). Proof. Let A(t) = [aij(t)] (i,j = l,...,n), C(t,u) = Fci(t,u)], i = l,...,n. Let X(t,t*) denote the fundamental solution of the homogeneous system dx/dt = A(t) x(t) which satisfies X(t*,t*) = I, the unit matrix, in other words the n columns of X are independent solutions of the homogeneous system with initial values at t* given by (1,0, o...,O),...,(0,...,0,1) respectively. Then we know from differential equation theory that any solution x(t) of the canonical system (5.3.1) satisfies the relation x(t) = X(t,t*)[x(t*) + ft X (Tt*)C(T,u(T))dT] Since (t*, x(t*) c P, P compact, and the vector functions and matrices C(t,u), X(t,t*) X-l(t,t*) are uniformly bounded for t,t* E [t,T], and u c U since U is compact, we conclude that x(t), tl < t < t2, with [tl,t2]c[t,T], admits of a uniform bound. Then, there is some constant N such that Ix(tl)l < N, Ix(t2)| 6

< N for all admissible pairs x(t), u(t), tl < t < t2. Also, A(t) x(t) + C(t, u(t)), tl < t < t2, admits of a uniform bound, say still N, and then ldx/dtJ < N, and the trajectories x(t), tl < t < t2, are uniformly Lipschitzian with constant N. Let us prove that the set B of all (t1,x(tl),t2,x(t2)), which are terminal points of trajectories, is closed. Let (t1,xl,t2,x2) cl Bo. Then there is a sequence of admissible pairs [Xk(t), k(t), tlk < t2k], k 1,2,..., and points t* such that tlk t, x (t lk) xl t2k t2, xk(t2k) + x2, t < t t < t < T, E P, uk(t) E U a.e. in [tlkt2k]. Since the trajectories xk(t), tlk - t < t2k, are equibounded, equicontinuous, and actually equilipschizian (of constant N and exponent 1), there is a subsequence, say still [xk], which converges in the metric p toward a continuous vector function x(t), tl < t < t2, which is necessarily Lipschitzian (of constant N) and hence AC in [tl,t2]. Also, P is compact, hence, we can extract the subsequence so that we have also (t*,xk(t*)) + (t*,x(t*)) for some t*, tl < t*< t2. It remains to prove that there is a measurable function u(t), tl < t < t2, with values u(t) E U, such that dx/dt = A(t) x(t) + C(t,u(t)) a.e. in [tl,t2], andthusx(t), u(t) is an admissible pair (with (tl,xl,t2,x2) E BO instead of (tl,xl,t2,x2) E B)). We shall denote by [tl,t2] the interval of definition of x(t), which is from now on a fixed interval. Let Q be the class of all B-measurable functions u(t), tl < t < t2, with [tl,t2] fixed as above, and values u(t) E U. For each u E ~, let y(t), tl < t < t2, be the AC n-vector function defined by the initial value differential problem

dy/dt = A(t) y(t) + C(t,u(t)) a.e. in [tl,t2], y(tl) = x(tl) Then Y(t2) = X(t2,tl)[y(tl) + ftz X-l(t,tl) C(tu(t))dt], where as usual X(ttl) denotes the fundamental solution of the homogeneous system dy/dt = A(t) y(t) with X(tl,tl) = I, the unit matrix. Let Q, denote the class of all n-vector functions v(t) = C(t,u(t)) with u E 2o. Let 22 denote the subset of En of all n-vectors z of the form Z = ft2 X(t,tl)C (t,u(t)) dt, u E. Finally, let 03 be the set of all n-vectors y(t2) defined as above for all u E ~. Obviously, we have a mapping u - v + z + y(t2), or o -+ *1 + a2 -+ 23. The class a has the property that, if u1,u2 cE and H is any Bmeasurable subset of [tl,t2], then the function u(t) = ul(t) if t E H, u(t) = u2(t) if t E [tl,t2]-H, is also an element of ~. Obviously, 21 possesses 0 the same property. By a version of a Lyapunsv's theorem [see App. E], the set 02 is convex and closed. (Then 02 is compact since Q2 is certainly bounded). Finally, 23 is the image of 22 by means of the linear transformation y(t2) = X(t2,tl) [x(tl) + z], and hence Q3 is also convex and closed (compact). Let us extend each vector uk(t), tlk < t < t2k, in the whole interval [t,T] by taking uk(t) = w, a fixed arbitrary point of U. Then let us restrict uk(t) to the fixed interval [tl,t2]. Now uk(t) is an element of Q, and its image in 23 is given by 8

Yk(t2) = X(t2,tl)[y(t2) + f t2 X- (t tl)C (t,uk(t))dt] while t2k k(t 2k)= X(t2k'tlk)[Xk (t,t)C(t,u(t))dt] lk Here tlk - tl, t2k + t2, x(tlk) + x (tl) tl) as k + a, as well as X(t2k,tlk) - X(t2,tl), and X-l(t,tlk) + X-l(t,tl) uniformly in [t,T]. Since xk(t ) + x(t2) = x2, we conclude that Yk(t2) has the same limit as k -+, or Yk(t2) + x(t2) = X2. Thus, x(t2) belongs to the closure of L3, and hence x(t2) belongs to 03 since L3 is closed. In other words, there is some u e 2 which generates x. Since g(tl,xl,t2,x2) is continuous on B, and hence on B nB and this set is not empty and compact, we conclude that the functional I[x,u]= g(tl,x(tl), t2,x(t2)) takes on both minimum and maximum in Q. Thereby (5.3.i) is proved. We can prove now, as a corollary of (5.3.i), an analogous statement for Lagrange problems. (5.3.ii) Existence Theorem (For Lagrange Problems and f and f Linear in x). Let A = [t,T] xE E, t, T finite, U a fixed compact set of the u-space Em, f = A(t) x + C(t,u), f = Ao(t) x + Co(t,u), A,Ao,C,C matrices with entries continuous in [t,T] and It,T] x U respectively. Let B be a closed subset of the t1xlt2x2-space. Let P be a compact subset of A and let Q be the class, which we suppose not empty, of all admissible pairs x,u whose trajectory x possesses at least one point (t*,x(t*)) E P. Then the functional (5.1.5) has an absolute minimum in L. If A = E1 x E then (5.5.i) still holds under the additional condition (c') of (3.4) (or alternate) (which here imply A (t) = O). As before, con9

ditions (c) and (c') hold here as a consequence of the hypotheses. Proof. If we introduce an auxiliary variable x satisfying the differential equation and initial condition dx0/dt = A (t)x(t) + C (t,u(t)), x0(tl) = O, then we can write the canonic system together with this equation in the form dx/dt = A(t)x(t) + C(t,u(t)), where x = (xx',..,xn) = (x,x), A = (A,A), C = (C,C), and then I[x,u] x0(t2). We can now apply theorem (5.3.i), where A is replaced by A1 = [t,T] x E+' B by B1 = B x [xl = 0] x El c E P by P1 = P x [x' = 0], gby 0) nfl, 2n+4t g(tl,x(tl),t2,x(t2)) = x (t2). This proves theorem (5.3.ii). 10

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