MATHEMATICS

V. N. KALININ

ON THE EXISTENCE OF AN ABSOLUTE MINIMUM IN ONE PROBLEM OF THE THEORY OF OPTIMAL PROCESSES

(Presented by Academician L. S. Pontryagin, 23 XI 1962)

In the present note an attempt is made to justify direct methods as applied to one problem in the theory of optimal control. Theorem 2 is given, adjacent to the known results of Tonelli in the classical calculus of variations \((^1)\), and establishing conditions for the existence of an absolute minimum in the problem posed and of a minimizing sequence converging to an element realizing this absolute minimum.

1. Definition 1. We shall call the class of admissible controls the set of all Lebesgue-measurable vector functions \(u(t)\), defined on the interval \(a=[0,\tau]\) and, for almost all \(t\in a\), mapping this interval into a closed domain \(G\) of the \(r\)-dimensional vector space \(R\). We denote the class of admissible controls by \(U\).

Definition 2. Suppose an operation \(A(u,x)\) is given, defined on the direct product of \(U\) and the space \(C_a\) of vector continuous functions with values in the \(n\)-dimensional vector space \(H\), defined on the interval \(a\), and having values in \(C_a\). If the indicated operation is bounded and, uniformly with respect to \(u\in U\), is a contraction operation in \(C_a\), then we shall say that it defines a controlled process. By a controlled process is meant the fixed point of the indicated operation for a given \(u\in U\), which is the solution of the functional equation

\[ x=A(u,x). \tag{1} \]

Let us combine all operations of the indicated kind into one class, denoted below by \(\Delta\). To each operation of this class there corresponds a single-valued and bounded mapping of the class of admissible controls \(U\) onto some set \(P\) of controlled processes in \(C_a\).

Definition 3. We shall call the operation that realizes the mapping \(U\to P\), corresponding to the operation \(A\in\Delta\), the resolvent of this operation, and denote it by \(S(u)\).

Thus, the class \(\Delta\) is isomorphic to some set of single-valued and bounded operations mapping the class of admissible controls \(U\) into the space \(C_a\).

Below the following problem is considered. Suppose a functional is given

\[ I(u)=\int_0^\tau f(x,u,t)\,dt, \tag{2} \]

where the function \(f(x,u,t)\) is understood to be such that the indicated integral exists (in the Lebesgue sense), is bounded and single-valued for any admissible control \(u\in U\) and the corresponding controlled process \(x\in P\) (with \(A\in\Delta\)). It is required to establish conditions under which, in the class of admissible controls \(U\), there exists a control \(u^*\) delivering to the functional (2) an absolute minimum in this class. We shall call this control optimal.

2. Definition 4. Let the integrand \(f(x,u,t)\) be defined for \(x\in H\), \(u\in G\), \(t\in a\) and in the indicated domain: a) together with the partial derivatives \(f_{u_\rho}(x,u,t)\) and \(f_{u_\rho u_\sigma}(x,u,t)\), \(\rho,\sigma=1,\ldots,r\), for almost all \(t\in a\) is continuous as a function of \(u\) and \(x\) and for all \(u\) and \(x\) (with \(|x|<\infty\))

measurable and almost everywhere on \(\alpha\) bounded with respect to \(t\); b) the real quadratic form

\[ \sum_{\rho,\sigma=1}^{r} f_{u_\rho u_\sigma}(x,u,t)\,\beta_\rho\beta_\sigma \]

is nonnegative for all \(x\in H,\ u\in G\) and for almost all \(t\in\alpha\). Then we shall say that the functional (2) is quasiregular in the class \(U\). If the metric in \(U\) is induced by its embedding in \(L^p\) \((1<p<\infty)\), then the following theorem can be proved:

Theorem 1. If the functional \(I(u)\) is quasiregular in the class of admissible controls \(U\) and the resolvent \(S(u)\) is strongly continuous for \(u\in U\), then \(I(u)\) is weakly lower semicontinuous on \(U\).

The proof of this theorem is based on considering the identity:

\[ \begin{aligned} I(u_0) = I(u_\nu) &+ \int_{0}^{\tau} [f(x_0,u_0,t)-f(x_\nu,u_0,t)]\,dt \\ &+ \int_{0}^{\tau} [f(x_\nu,u_0,t)-f(x_\nu,u_\nu,t)]\,dt, \end{aligned} \tag{3} \]

where the sequence \(\{u_\nu\}\in U\) converges weakly to \(u_0\in U\), \(x_\nu=S(u_\nu)\), \(x_0=S(u_0)\), and on estimating the integrals occurring on the right-hand side. It can be shown that, under the assumptions of the theorem,

\[ I(u_0)\leq \lim_{\nu\to\infty} I(u), \]

which also means the weak lower semicontinuity of \(I(u)\) on \(U\).

1. Denote by \(\Delta_0\) the set of operations from \(\Delta\) having a strongly continuous resolvent. Since \(U\in L^p\) for \(1<p<\infty\) is evidently bounded and weakly closed and, by virtue of the regularity of \(L^p\) \((1<p<\infty)\), weakly compact (the Gantmacher–Shmulyan–Eberlein theorem \((^2)\)), it follows, taking into account Theorem 1 and known theorems of nonlinear functional analysis \((^2)\), that

Theorem 2. If \(A\in\Delta_0\) and the functional \(I(u)\) (2) is quasiregular in \(U\), then in the class of admissible controls there exists a control \(u^*\) giving it an absolute minimum in this class (an optimal control).

Hence follows the existence in the class \(U\) of a minimizing sequence converging to \(u^*\) at least weakly. As for the set \(\Delta_0\), the following theorem is useful for singling it out from the class \(\Delta\).

Theorem 3. In order that \(A\in\Delta_0\), it is sufficient that \(A\in\Delta\) and that, for every \(x\in C_\alpha\), \(A\) be strongly continuous with respect to \(u\) on \(U\).

We note that from the condition of the theorem there follows, generally speaking, not only the strong continuity of the resolvent \(S(u)\), but also its complete continuity. Indeed, by the theorem of Tsitlanadze \((^3)\), a strongly continuous operation defined in a space with a weakly compact ball is completely continuous; thus the following is established:

Corollary. If the operation \(A\in\Delta_0\), then its resolvent \(S(u)\) is completely continuous.

Moreover, evidently, the class \(P\) of controlled processes in \(C_\alpha\) is compact, i.e. it is a set of uniformly bounded and equicontinuous functions.

In conclusion, the author expresses deep gratitude to V. I. Zubov for valuable comments made in the discussion of the present work.

Received
23 XI 1962

References

1. L. Tonelli, Fondamenti di calcolo delle variazioni, Bologna, 1921—1923.
2. M. M. Vainberg, Variational Methods for the Study of Nonlinear Operators, 1956.
3. E. S. Tsitlanadze, Mat. sbornik, 29 (71), no. 1, 3 (1951).