UDC 519.3

F. P. VASIL'EV, R. P. IVANOV (Bulgaria)

SOME APPROXIMATE METHODS FOR SOLVING TIME-OPTIMAL PROBLEMS IN BANACH SPACES IN THE PRESENCE OF PHASE CONSTRAINTS

(Presented by Academician A. N. Tikhonov on 15 IV 1970)

1. Statement of the problem.

Let \(U\) be a given subset of a topological linear space \(Z\); the elements \(u \in U\) will be called controls. Let \(t\) be time, and \(t_0\) a known initial instant of time. Suppose that for each \(t \geq t_0\) there is given a set \(\Omega[t_0,t]\) of functions \(x(\tau)\), defined for \(t_0 \leq \tau \leq t\) and taking their values in a given Banach space \(B\), such that \(\Omega[t_0,t] \subset \Omega[t_0,t']\) for all \(t_0 \leq t' < t\). A function \(x(\tau) \in \Omega[t_0,t]\) will be called a possible trajectory. Suppose that for all \(t \geq t_0\) there is given a mapping that assigns to each \(u \in U\) a possible trajectory \(x(\tau,u) \in \Omega[t_0,t]\). Let \(G(t)\), \(t \geq t_0\), be given subsets of \(B\), and let \(I_\alpha(x(\tau),u,t)\) \((\alpha \in A\), a known set of indices) be functionals defined on the topological product \(\Omega[t_0,t] \times U \times [t \geq t_0]\). We shall say that the trajectory \(x(\tau,u) \in \Omega[t_0,t]\) satisfies the phase constraints if \(x(\tau,u) \in G(\tau)\) for \(t_0 \leq \tau \leq t\), \(I_\alpha(x(\tau,u),u,t) \leq 0\) \((\alpha \in A)\), and such a trajectory will be called admissible on \(t_0 \leq \tau \leq t\). The subset of those \(u \in U\) for which the corresponding trajectory \(x(\tau,u)\) is admissible on \(t_0 \leq \tau \leq t\) will be denoted by \(U(t)\); the set of those \(x \in B\) for which there exists \(u \in U(t)\) such that \(x(t,u)=x\) will be denoted by \(X(t)\). Finally, let \(Y(t)\), \(t \geq t_0\), be given subsets of \(B\). The time-optimal problem consists in finding such \(T^*\) and \(u^* \in U(T^*)\) that \(x(T^*,u^*) \in Y(T^*)\), while for all other \(T\) and \(u \in U(T)\) for which \(x(T,u) \in Y(T)\), the inequality \(T \geq T^*\) holds. Such an instant \(T^*\) and control \(u^* \in U(T^*)\) will be called optimal.

Everywhere below we shall assume that the following conditions, which we shall call conditions A, are satisfied: 1) the set \(U\) is convex and bicompact in the topology of the space \(Z\); 2)

\[ x(\tau,\alpha u+(1-\alpha)v) \equiv \alpha x(\tau,u)+(1-\alpha)x(\tau,v), \]

\(t_0 \leq \tau \leq t\), for all \(u,v \in U\), \(0 \leq \alpha \leq 1\), and \(t \geq t_0\); 3) for any net \(\{u_k\} \subset U\) converging to \(u\) in \(Z\) ((\(^1\)), p. 29), the net \(\{x(\tau,u_k)\} \subset B\) converges to \(x(\tau,u)\) in the weak topology of \(B\) for all \(\tau \geq t_0\); 4) the sets \(G(t)\), \(t \geq t_0\), are convex and closed in \(B\); 5) the functionals \(I_\alpha(x(\tau),u,t)\) are convex jointly in \((x(\tau),u) \in \Omega[t_0,t] \times U\), and \(I_\alpha(x(\tau,u),u,t)\) are lower semicontinuous in \(u \in U\) ((\(^1\)), p. 27); 6) the sets \(Y(t)\), \(t \geq t_0\), are convex and weakly bicompact in \(B\) for all \(t \geq t_0\).

In describing and investigating the convergence of the methods given below for solving the time-optimal problem we shall need more stringent conditions, which we shall call conditions B: 1) conditions A are satisfied; 2)

\[ \sup_{u \in U}\|x(t+\Delta t,u)-x(t,u)\|_B \to 0 \]

as \(\Delta t \to 0\); 3) for any sequence \(t_k\), \(t_k \leq t\), \(t_k \to t\), and any \(x_k \in G(t_k)\) such that \(\|x-x_k\|_B \to 0\) \((k \to \infty)\), one has \(x \in G(t)\); 4) if \(I_\alpha(x(\tau,u),u,t) \leq 0\), then \(I_\alpha(x(\tau,u),u,t') \leq 0\) for all \(t_0 \leq t' < t\), and also

\[ \lim_{\Delta \to +0} I_\alpha(x(\tau,u),u,t-\Delta t) \geq I_\alpha(x(\tau,u),u,t) \]

\((\alpha \in A)\); 5)

\[ \sup_{y \in Y(t)} \min_{z \in Y(\tau)} \|y-z\| \to 0 \]

as \(\tau \to t-0\), and

\[ \sup_{\substack{z \in Y(\tau)\\ y \in Y(t)}} \min \|y-z\| \to 0 \]

as \(\tau \to t\).

2. Controllability and optimality criteria.

For

in the study of the time-optimal problem, an important role is played by the functional

\[ M(c,t)=\min_{x\in X(t)}\min_{y\in Y(t)}(c,x-y) =\min_{u\in U(t)}(c,x(t,u))-\max_{y\in Y(t)}(c,y), \]

where \(c\in B^*\) is the space dual to \(B\), and \((c,z)\) is the value of the linear functional \(c\) on the element \(z\in B\). Under conditions A, the set \(X(t)\) is convex and weakly bicompact in \(B\), and therefore \(M(c,t)\) is defined for all \(c\in B^*\), \(t\ge t_0\).

Definition. We shall call the system \(\{x(t,u)\}\) \(U(T)\)-controllable if there exists a \(u\in U(T)\) such that \(x(T,u)\in Y(T)\); otherwise the system is \(U(T)\)-uncontrollable.

Theorem 1. If conditions A are satisfied, then: 1) the system \(\{x(t,u)\}\) is \(U(T)\)-controllable if and only if

\[ M(c,T)=\min_{x\in X(T)}(c,x)-\max_{y\in Y(T)}(c,y) \]

for all \(c\in B^*\); 2) \(T^*\) is optimal if and only if \(M(c,t)\le 0\) for all \(c\in B^*\), and for any \(t_0\le t<T^*\) there exists a \(c_t\in B^*\) such that \(M(c_t,t)>0\).

Lemma 1. Let conditions A be satisfied. Then, for any \(t\ge t_0\), the functional \(M(c,t)\): 1) is concave in \(c\); 2) \(M(\alpha c,t)=\alpha M(c,t)\), \(\alpha=\mathrm{const}\ge 0\); 3) satisfies a Lipschitz condition in \(c\) in the norm of \(B^*\); 4) is upper semicontinuous in \(c\) in the \(B\)-topology of the space \(B^*\).

Lemma 2. Let conditions B be satisfied. Then: 1) \(M(c,t)\), for any fixed \(c\in B^*\), is lower semicontinuous in \(t\) and continuous from the left in \(t\); 2) if, moreover, the system \(\{x(t,u)\}\) is \(U(T)\)-controllable and \(M(c,s)>0\) for some \(c\in B^*\), \(t_0\le s<T\), then there exists a time \(t\), \(s<t\le T\), such that \(M(c,t)=0\), \(M(c,\tau)>0\) for \(s\le \tau<t\).

3. Method I. To describe this method, introduce the functional

\[ \rho(t)=\max_{\|c\|\le 1} M(c,t). \]

From Lemma 1 and the bicompactness of the unit ball in \(B^*\) in the \(B\)-topology of \(B^*\) (Alaoglu’s theorem \((^2)\), p. 459), it follows that \(\rho(t)\) is defined for all \(t\ge t_0\).

Theorem 2. If conditions A are satisfied, then: 1) the system \(\{x(t,u)\}\) is \(U(T)\)-controllable if and only if \(\rho(T)=0\); 2) \(T^*\) is optimal if and only if \(\rho(T^*)=0\), \(\rho(t)>0\) for \(t_0\le t<T^*\).

Lemma 3. If conditions B are satisfied, then \(\rho(t)\) is lower semicontinuous for \(t\ge t_0\) and continuous from the left for \(t>t_0\).

We shall describe Method I under the assumption that condition B is satisfied and the system \(\{x(t,u)\}\) is \(U(T)\)-controllable. Specify some sequence \(\{\delta_k\}_{k=1}^{\infty}\), \(\delta_k\ge 0\), \(\delta_k\to 0\) \((k\to\infty)\), a constant \(R>0\), and a continuous function \(\alpha(\rho)\), \(\alpha(0)=0\), \(0<\alpha(\rho)\le R\rho\) for \(\rho>0\). As the initial approximation, take \(t_0\) and such a \(c_0\in B^*\), \(\|c_0\|\le R\), that \(\alpha(\rho(t_0))\le M(c_0,t_0)\) (naturally, one should assume \(\rho(t_0)>0\)). Suppose the \((k-1)\)-st approximation \(t_{k-1},c_{k-1}\) is known and \(t_0<t_1<\cdots<t_{k-1}<T\), \(\rho(t)>0\) for \(t_0\le t\le t_{k-1}\), \(0<\alpha(\rho(t_{k-1}))\le M(c_{k-1},t_{k-1})\), \(\|c_{k-1}\|\le R\). Then determine \(t_k\) from the conditions \(t_{k-1}<t_k\le T\), \(M(c_{k-1},t_k)\le\delta_k\), \(M(c_{k-1},\tau)>0\) for \(t_{k-1}\le \tau<t_k\) (the existence of such a \(t\) follows from Lemma 3). By the choice of \(t_k\), \(\rho(t)>0\) for \(t_{k-1}\le t<t_k\). Next, find \(c_k\) from the conditions \(\|c_k\|\le R\), \(\alpha(\rho(t_k))\le M(c_k,t_k)\). If \(M(c_k,t_k)=0\), then \(\alpha(\rho(t_k))=0\), \(\rho(t_k)=0\); the time \(t_k\) is optimal and the process is finished. If \(M(c_k,t_k)>0\), then \(\rho(t_k)>0\), and the process is continued. If this process does not terminate after a finite number of steps, then from Lemmas 1–3 and Theorem 2 it follows that

Theorem 3. Let condition B be satisfied and the system \(\{x(t,u)\}\) be \(U(T)\)-controllable. Then the sequence \(t_k\) converges to the optimal \(T^*\).

4. Method II. To describe Method II, introduce the functional

\[ \chi(t)=\sup_{c\in \Pi(t)} M(c,t), \]

where

\[ \Pi(t)=\{c:c\in B^*,\ \max_{y\in Y(t)}(c,y)=-1\}. \]

If conditions A are satisfied and \(0\notin Y(t)\), \(0\in X(t)\), then \(\Pi(t)\) is nonempty and \(\chi(t)>-\infty\); if \(0\in Y(t)\), then \(\Pi(t)\) is empty and, by definition, we put \(\chi(t)=-\infty\).

Theorem 4. Suppose conditions A are satisfied, and suppose \(0\in X(t)\) for all \(t\ge t_0\). Then: 1) the system \(\{x(t,u)\}\) is \(U(T)\)-controllable if and only if

when \(\chi(T)\le 0\); 2) \(T^*\) is optimal if and only if \(\chi(T^*)\le 0\), \(\chi(t)>0\) for \(t_0\le t<T^*\).

Lemma 4. If condition B is satisfied and \(0\in X(\tau)\) for all \(\tau\) from a sufficiently small neighborhood of \(t\), then \(\chi(t)\) is lower semicontinuous at the point \(t\). If, moreover, \(0\notin Y(t)\) and, for some \(\omega_t>0\), one of the sets \(X(t)\cap[\omega_tY(t)]^0\) or \([X(t)]^0\cap[\omega_tY(t)]^*\) is nonempty, then \(\chi(t)\) is left-continuous at the point \(t>t_0\).

We describe method II under the assumption that condition B is satisfied, the system \(\{x(t,u)\}\) is \(U(T)\)-controllable, and \(0\in X(t)\) for all \(t_0\le t\le T\). Let a sequence \(\delta_k\ge 0\) \((k=1,2,\ldots)\), \(\delta_k\to 0\) \((k\to\infty)\), and a function \(\alpha(\chi)\), \(\alpha(\chi)<\chi\) for all \(\chi\), \(\alpha(\chi)>0\) for \(\chi>0\), be given. As the initial approximation take \(t_0\) and \(c_0\in\Pi(t_0)\), \(M(c_0,t_0)\ge \alpha(\chi(t_0))>0\). Suppose the \((k-1)\)-st approximation \(t_{k-1}\), \(c_{k-1}\in\Pi(t_{k-1})\), \(t_0<t_1<\cdots<t_{k-1}<T\), \(\chi(t)>0\) for \(t_0\le t\le t_{k-1}\), \(0<\alpha(\chi(t_{k-1}))\le M(c_{k-1},t_{k-1})\) \((k\ge 1)\), is known. Then, by Lemma 3, there is a \(t_k\), \(t_{k-1}<t_k\le T\), such that \(M(c_{k-1},t_k)\le\delta_k\), \(M(c_{k-1},t)>0\) for \(t_{k-1}\le t<t_k\). Then \(\chi(t)>0\) for \(t_{k-1}\le t<t_k\). It is possible that \(0\in Y(t_k)\); then \(t_k\) is the optimal time and the process is finished. If \(0\notin Y(t_k)\), then we find \(c_k\in\Pi(t_k)\) from the condition \(\alpha(\chi(t_k))\le M(c_k,t_k)\). If \(M(c_k,t_k)\le 0\), then \(\chi(t_k)\le 0\) and \(t_k\) is optimal; the process is finished. If \(M(c_k,t_k)>0\), then \(\chi(t_k)>0\) and we continue the process. If this process does not end in a finite number of steps, then Lemma 4 and Theorem 4 imply

Theorem 5. Suppose condition B is satisfied, the system \(\{x(t,u)\}\) is \(U(T)\)-controllable, \(0\in X(t)\) for \(t_0\le t\le T\), and suppose that for some \(\omega_t>0\) one of the sets \(X(t)\cap[\omega_tY(t)]^0\) or \([X(t)]^0\cap[\omega_tY(t)]^*\) is nonempty for \(t_0<t\le T\). Then the sequence \(t_k\) \((k=1,2,\ldots)\), obtained by method II, converges to the optimal \(T^*\).

In practice, apparently, instead of \(\chi(t)\) it is more convenient to work with the functional

\[ \omega(t)=\inf_{c\in\Pi(t)}\max_{x\in X(t)}(-c,x). \]

Since \(\chi(t)=1-\omega(t)\), it is not difficult to set out method II and formulate Theorems 4, 5 and Lemma 4 using \(\omega(t)\).

5. Applications. Example 1. Let the process be described by the system

\[ \dot{x}(t)=Ax(t)+Bu(t),\qquad t\ge 0,\qquad x(0)\in X_0, \]

where \(x=(x^1,\ldots,x^n)\), \(u=(u^1,\ldots,u^r)\), \(A\) and \(B\) are constant matrices of dimensions \(n\times n\) and \(n\times r\), respectively; \(X_0\) is a given subset of the Euclidean space \(E_n\). Take \(U\): either

\[ U_1=\{u(t):\operatorname{vrai\,sup}_{0\le t\le T}|u^i(t)|\le \alpha_i=\mathrm{const},\quad i=1,\ldots,r\}, \]

or

\[ U_2=\left\{u(t):\int_0^T u^2(t)\,dt\le \alpha=\mathrm{const}\right\}, \]

or \(U_3=U_1\cap U_2\). It is required to transfer the system in minimum time from the set \(X_0\) to the set \(Y(t)\), observing the phase constraints:

\[ x(t)\in G(t),\quad t\ge 0,\qquad \int_0^T x^2(t)\,dt\le \beta=\mathrm{const}; \]

here \(Y(t)\), \(G(t)\) are given subsets of \(E_n\).

Theorem 6. Suppose 1) the system is \(U(T)\)-controllable for some \(T<\infty\); 2) \(X_0\) is convex, closed, bounded; 3) \(G(t)\) is convex, closed and \(G(t-0)\subset G(t)\) for all \(t\ge 0\); 4) \(Y(t)\) is convex, closed, bounded, left-continuous in the Hausdorff sense, and \(Y(t+0)\subset Y(t)\), \(t\ge 0\). Then the sequence \(\{t_k\}\) from method I converges to the optimal \(T^*\). If, in addition to 1)—4), the following conditions are satisfied: 5) the rank of the matrix \(\{B,AB,\ldots,A^{n-1}B\}\) is equal to \(n\); 6) for any \(T>0\) there exists a ball \(K_T\subset E_n\) with center at zero such that \(K_T\subset G(t)\) for \(t_0\le t\le T\), then the sequence \(\{t_k\}\) from method II converges to the optimal \(T^*\).

\[ \text{* Here }[\,]^0\text{ denotes the interior of a set.} \]

Some variants of methods I, II have been studied, for example, in \((^{3-6})\).

Example 2. Let the process be described by the conditions:

\[ x_t=x_{ss},\quad x\equiv x(s,t),\quad (s,t)\in Q_T=\{0\leq s\leq 1,\ 0\leq t\leq T\}, \]

\[ x_s(0,t)=0,\quad x_s(1,t)=\alpha[u(t)-x(1,t)],\quad x(s,0)=0,\quad \alpha=\mathrm{const}>0. \]

From these conditions, for each fixed \(u=u(t)\in U(T)=\{u(t): u(t)\in L_2[0,T],\ \operatorname{vrai\,sup}_{0\leq t<T}|u(t)|\leq 1\}\), \(x=x(s,t,u)\) is uniquely determined \((^7)\). It is required, in the minimal time \(T\), to achieve the fulfillment of

\[ x(s,t,u)\in Y=\left\{y(s): y(s)\in L_2[0,1],\int_0^1 |y(s)-y_0(s)|^2\,ds\leq \delta^2\right\},\quad \delta=\mathrm{const}>0, \]

where \(y_0(s)\) is a given function from \(L_2[0,1]\). Here one must take

\[ M(c,t)=\min_{u\in U(T)}\min_{y\in Y}\int_0^1 c(s)[x(s,t,u)-y(s)]\,ds,\quad c(s)\in L_2[0,1]. \]

Theorem 7. If, for some \(T<\infty\), the system is \(U(T)\)-controllable, then the sequence \(\{t_k\}\), determined by method I or II, converges to the optimal \(T\) (method II cf. \((^8)\), pp. 303, 380; for method I see \((^9)\)).

Example 3. Let the process be described by the conditions:

\[ x_{tt}=a^2x_{ss}+u_0(s,t),\quad x\equiv x(s,t),\quad (s,t)\in Q_T\quad (a=\mathrm{const}>0), \]

\[ x(0,t)=x(1,t)=0,\quad x(s,0)=u_1(s),\quad x_t(s,0)=u_2(s). \]

From these conditions, for each \(u=(u_0,u_1,u_2)\in U=U_0\times U_1\times U_2\),

\[ U_0=\{u_0(s,t): \|u_0\|_{L_2(Q_T)}\leq \alpha_0\},\quad U_1=\{u_1(s): \|u_1\|_{W_2^{(2)}[0,1]}\leq \alpha_1,\ u_1(0)=u_1(1)=0\}, \]

\[ U_2=\{u_2(s): \|u_2\|_{W_2^{(1)}[0,1]}\leq \alpha_2,\ u_2(0)=u_2(1)=0\},\quad \alpha_i=\mathrm{const}\geq 0,\ i=0,1,2, \]

the solution \(x=x(s,t,u)\in W_2^{(2)}(Q_T)\) is uniquely determined \((^{10})\). It is required, in the minimal time \(T\), to achieve the fulfillment of

\[ \int_0^1 \left(|x(s,T,u)-\bar y_0(s)|^2+|x_t(s,T,u)-\bar y_1(s)|^2\right)\,ds\leq \delta, \]

while observing the phase constraints

\[ \beta_1\int_0^1 |x(s,T,u)-\bar y_2(s)|^2\,ds+ \beta_2\int_0^1 |x_t(s,t,u)-\bar y_3(s)|^2\,ds\leq \beta_3, \]

where \(\bar y_i(s)\in L_2[0,1]\) are given functions, \(\delta=\mathrm{const}\geq 0\), \(\beta_i=\mathrm{const}\geq 0\). Let us take

\[ c=(c_1(s),c_2(s)),\quad \|c(s)\|=\|c\|_{L_2^{(2)}[0,1]}= \left(\int_0^1 (|c_1(s)|^2+|c_2(s)|^2)\,ds\right)^{1/2},\quad y=(y_1(s),y_2(s)); \]

\[ \bar y=(\bar y_0(s),\bar y_1(s)),\quad Y=\{y:\ \|y-\bar y\|_{L_2^{(2)}[0,1]}\leq \delta\}, \]

\[ M(c,t)=\min_{u\in U(t)}\min_{y\in Y}\int_0^1\bigl(c_1(s)x(s,t,u)+c_2(s)x_t(s,t,u)-c_1(s)\bar y_1(s)- \]

\[ -c_2(s)\bar y_2(s)\bigr)\,ds. \]

Theorem 8. If, for some \(T<\infty\), the system is \(U(T)\)-controllable, then the sequence \(\{t_k\}\) from method I converges to the optimal \(T^*\).

Moscow State University
named after M. V. Lomonosov

Received
13 IV 1970

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