MATHEMATICS

Yu. V. EGOROV

OPTIMAL CONTROL IN A BANACH SPACE

(Presented by Academician L. S. Pontryagin on 6 XII 1962)

In a Banach space an equation and initial conditions are given:

\[ \frac{dx(t)}{dt}=f(x(t),u(t)),\qquad x(a)=x_0\quad (a\leq t\leq b), \tag{1} \]

where \(u(t)\)—the “control”—is a measurable function with values in a given set \(U\) of some topological space. This control is to be chosen so as to minimize a functional of the form

\[ \int_0^b f^0(x(t),u(t))\,dt, \]

and certain conditions are imposed on \(x(b)\) (for example, \(x(b)=x_1\), or \(x(b)\in S\), where \(S\) is a given manifold, or \(x(b)\in T\), where \(T\) is a convex body). For a finite-dimensional space such problems were studied in \((^1)\). Under certain restrictions on \(f(x,u)\) we obtain necessary conditions for optimality of the control in the form of Pontryagin’s maximum principle.

1. Let \(x(b)=x_1\) be a given point, but the time instant \(b\) is not fixed. It is assumed that problem (1), for fixed \(u(t)\), has a unique solution \(x(t)\), which lies in \(B_1\) for almost all \(t\) in \((a,b)\), and \(f(x,u)\in B_2\) if \(x\in B_1\) and \(u\in U\) \((B_1\subset B_2)\); \(\varphi(x,u)\) and \(\partial\varphi(x,u)/\partial x\) are continuous in \(x\) and \(u\). Denote

\[ x^0(t)=\int_a^t f^0(x(\xi),u(\xi))\,d\xi,\quad y=(x^0;x),\quad \varphi=(f^0;f), \]

\[ A_1=R^1\times B_1,\quad A_2=R^1\times B_2. \]

If, for \(t_1\leq t\leq t_2\), \(u(t)\) does not depend on \(t\), then \(x(t)\)—the solution of (1)—is continuous on this interval in the norm of \(B_1\).

\(i_1)\) Let \(\tilde u_1(t)=u(t)\) for \(a<t<\tau-\varepsilon\) and \(\tau<t<b\), \(\tilde u_1(t)=v\in U\) if \(\tau-\varepsilon<t<\tau\); \(\tilde u_2(t)=u(t)\) for \(a<t<\tau-\varepsilon\), \(\tilde u_2(t)=u(t+\varepsilon)\) when \(\tau-\varepsilon<t<b-\varepsilon\); \(\tilde u_3(t)=u(t)\) for \(a<t<\tau\), \(\tilde u_3(t)=u(\tau)\) for \(\tau<t<\tau+\varepsilon\), \(\tilde u_3(t)=u(t-\varepsilon)\) if \(\tau+\varepsilon<t<b+\varepsilon\). Let \(\tilde x_i(t)\) be the solution of (1) for \(u(t)=\tilde u_i(t)\). Put \(\theta_1(t,\tau)\equiv 0\); \(\theta_2(t,\tau)=0\) for \(t<\tau\), \(\theta_2(t,\tau)=-1\) for \(t\geq\tau\); \(\theta_3(t,\tau)=-\theta_2(t,\tau)\).

There exists a set \(M\) of full measure on \((a,b)\) of points \(\tau\), for which there exists

\[ \lim_{\varepsilon\to 0}\varepsilon^{-1}\bigl[\tilde y_i(t+\varepsilon\theta_i(t,\tau))-y_i(t)\bigr]=z_i(t)\in A_2. \]

Moreover,

\[ \lim_{\varepsilon\to 0}\|\tilde x_i(b+\varepsilon\theta_i(b,\tau))-x(b)\|_{B_1}=0\quad (i=1,2,3). \]

\(i_2)\) Let \(a^*\in A_2^*\) (i.e., \(a^*\) is a linear continuous functional on \(A_2\); the value of \(a^*\) at \(y\in A_2\) is denoted by \((a^*,y)\)). There exists a function \(\psi(t)\) such that \(\psi(b)=a^*\); for almost all \(t\) in \((a,b)\), \(\psi(t)\in A_2^*\); \((\psi(t),z_i(t))=\mathrm{const}\) \((i=1,2,3;\ \tau\leq t\leq b)\), and, if \(\psi(t)=(\psi_0(t),\psi_1(t))\), where \(\psi_1(t)\in B_2^*\), then \(\psi_0(t)=\mathrm{const}\).

\(i_3)\) The reserve of controls \(U\) is sufficiently large. More precisely, there exist \(\rho>0\) and a set \(\omega\subset U\) such that, for \(x\) in the ball \(\|x-x_1\|_{B_1}\leq\rho\), the set \(\{f(x,u)\}_{u\in\omega}\) is open and homeomorphic to \(\omega\).

Theorem 1. If conditions \(i_1)\), \(i_2)\), \(i_3)\) are fulfilled, then for every optimal control \(u(t)\) there exists \(a^*\in A_2^*\) \((a^*\ne0)\) such that every function \(\psi(t)\) from \(i_2)\) has the property that the function

\[ H(\psi(t),x(t),u)=(\psi(t),\varphi(x(t),u)) \tag{2} \]

of the variable \(u\in U\) almost everywhere on the interval \(a\leq t\leq b\) attains a maximum at the point \(u=u(t)\). (Here \(x(t)\) is the solution of (1) corresponding to \(u(t)\).) Moreover, \(H(\psi(t),x(t),u(t))\equiv 0,\ \psi_0\leq 0\).

Remark 1. If \(f(x,u)\) is a bounded operator, \(B_1=B_2\), then \(z(t)\) and \(\psi(t)\) satisfy the equations

\[ \frac{dz(t)}{dt}=\frac{\partial\varphi(x(t),u(t))}{\partial x}\,z(t),\qquad \frac{d\psi(t)}{dt}=-\left(\frac{\partial\varphi(x(t),u(t))}{\partial x}\right)^*\psi(t). \]

If, however, \(f\) is an unbounded operator, \(z(t)\) and \(\psi(t)\) satisfy these equations only in a certain generalized sense.

Remark 2. The essential nature of requirement \(i_3\) in the case of an infinite-dimensional space is shown by the following example.

If in \(l_2\) the problem is posed: minimize the time \(b-a\) (i.e. \(f^0\equiv 1\)) under the conditions

\[ \frac{dx}{dt}=u,\qquad x_n(a)=0,\qquad x_n(b)=\frac1n, \]

\[ U=\left\{u:\ |u_n|\leq \frac1n+\frac1{n^2}\right\}\qquad (n=1,2,\ldots), \]

then, as is easy to see, the control \(u(t)=\left(1,\frac12,\ldots,\frac1n,\ldots\right)\) is optimal, and there does not exist a vector \(\psi(t)\) satisfying the conditions of the maximum principle. If, however, \(x\) and \(u\) are regarded as elements of a space in whose topology the set \(U\) has an interior point, such a vector \(\psi(t)\) always exists.

1. Let \(S\) be a smooth manifold in \(B_1\), and suppose that in the optimal-control problem it is required that \(x(b)\in S\) (the time \(b\) is not fixed).

\(j_1\)) Let \(T_1\) be the tangent manifold to \(S\) at the point \(x(b)\), and \(T_2=B_2/T_1\). Denote by \(y^\pi\) the class in \(T_2\) in which \(y\in B_2\) lies. There exist \(\rho>0\) and a set \(\omega\subset U\) such that, for every \(x\) in the ball \(\|x-x(b)\|_{B_1}\leq \rho\), the set \(\{f^\pi(x,u)\}_{u\in\omega}\) is open in \(T_2\) and homeomorphic to \(\omega\).

Theorem 2. If conditions \(i_1), i_2), j_1)\) are fulfilled, then for every optimal control \(u(t)\) there exists \(a^*\in A_2^*\) \((a^*\neq 0)\), satisfying the transversality conditions (i.e. \((a^*,y)=0\) for \(y\in T_1\)) and such that, whatever the function \(\psi(t)\) from condition \(i_2)\) may be, the function (2) attains a maximum with respect to \(u\) at the point \(u=u(t)\). Moreover, \(H(\psi(t),x(t),u(t))\equiv 0,\ \psi_0\leq 0\).

Remark 1. The theorem is also valid in the case when \(x(a)\) is not fixed, but merely belongs to a given manifold \(S_0\). In this case \(\psi(t)\) satisfies the transversality conditions also for \(t=a\).

Remark 2. Theorem 2 makes it possible to consider equations with coefficients depending on \(t\), problems with fixed \(b\), etc. (see \([1]\)).

Example 1. \(dx(t)/dt=A(t)x(t)+u(t),\ x(a)=x_0,\ x(b)=x_1,\ f^0=1,\ U\) is the ball of unit radius in the space \(B\), strictly convex (i.e., if \(\|u_1\|+\|u_2\|=1\), then \(\|u_1+u_2\|<2\), if \(u_1\neq u_2\)), and \(A(t)\) is a bounded linear operator in this space. From Theorem 2 follows the uniqueness of the optimal control.

Example 2. Let

\[ \mathcal L x(t,s)=\sum_{i,j=1}^{n} a_{ij}(t,s)\frac{\partial^2 x(t,s)}{\partial s_i\partial s_j} +\sum_{i=1}^{n} b_i(t,s)\frac{\partial x(t,s)}{\partial s_i} +c(t,s)x(t,s) \tag{3} \]

be an elliptic operator with sufficiently smooth coefficients, \(s=(s_1,\ldots,s_n)\in\Omega\), where \(\Omega\) is a bounded domain with smooth boundary \(\Gamma\) in \(n\)-dimensional space, \(a\leq t\leq b\).

Given the equation \(\partial x(t,s)/\partial t=\mathcal Lx(t,s)+f(t,s)+u(t,s)\) and the conditions:
\(x(a,s)=x_0(s)\in W_2^{(2)}(\Omega),\ x(b,s)=x_1(s)\in W_2^{(2)}(\Omega),\ x(t,s)|_\Gamma=0\) (see \([4]\)),

where

\[ U=\left\{u(t,s):\int_{\Omega}u^2(t,s)\,ds\leq 1\right\}. \]

It is required to minimize the time \(b-a\) of transition from \(x_0(s)\) to \(x_1(s)\), i.e. \(f^0(x,u)\equiv 1\). It is not difficult to show the existence of an optimal control (provided there is at least one \(u(t,s)\) for which there exists a solution with the given boundary conditions), and its uniqueness follows from Theorem 2 (see \((2)\)).

1. Finally, consider the optimal-control problem when \(x(b)\) belongs to a given convex body in a Banach space, for example,
   \[ \|x(b)-x_1\|_B\leq \rho. \]
   The time \(b\) is not fixed.

\(k_1)\) Let \(A=R^1\times B\). For almost all \(t\), the solution of problem (1) satisfies \(x(t)\in B\) and \(f(x(t),u(t))\in B\).

\(k_2)\) Let \(\tilde u(t)\ne u(t)\) only when \(\tau-\varepsilon<t<\tau\), and let \(\tilde u(t)=v\) for these \(t\) (\(v\) is an arbitrary element of \(U\)), and let \(\tilde x(t)\) be the solution of (1) corresponding to \(\tilde u(t)\). On the interval \((a,b)\) one can specify a set \(M\) of full measure of points \(\tau\) for which

\[ \lim_{\varepsilon\to0}\varepsilon^{-1}[\tilde y(t)-y(t)]=z(t)\in A. \]

\(k_3)\) Let \(a^*\in A^*\) be an arbitrary continuous linear functional on \(A\). There exists a function \(\psi(t)\in A^*\) \((a\leq t\leq b)\) such that \(\psi(b)=a^*\), \((\psi(t),z(t))=\operatorname{const}\) \((\tau\leq t\leq b)\), and \(\psi_0(t)=\operatorname{const}\).

\(k_4)\) For every \(\tau\in M\) there exist

\[ \lim_{\varepsilon\to0}\varepsilon^{-1}\left[\tilde y_i\bigl(t+\varepsilon\theta_i(t,\tau)\bigr)-y_i(t)\right]=z_i(t)\in A \quad (i=2,3), \]

where \(\tilde y_i(t)\) and \(\theta_i(t,\tau)\) are defined in \(i_1\). Condition \(k_3)\) is fulfilled if \(z(t)\) is replaced by \(z_2(t)\) or \(z_3(t)\).

Theorem 3. If conditions \(k_1), k_2), k_3)\) are fulfilled, then there exists \(a^*\in A^*\) \((a^*\ne 0)\) such that the function of \(u\)

\[ H(\psi(t),x(t),u)=(\psi(t),\varphi(x(t),u)) \]

attains its maximum at \(u=u(t)\), where \(u(t)\) is the optimal control and \(x(t)\) is the corresponding solution of problem (1). In this case \(\psi_0\leq 0\). If condition \(k_4)\) is also fulfilled, then \(H(\psi(t),x(t),u(t))\equiv 0\).

Examples. If the function \(f^0(x,u)=1\), then Theorem 3 implies the uniqueness of the optimal control for the following problems:

\[ 1)\quad \frac{\partial x(t,s)}{\partial t} = \sum_{k=1}^{n} A_k(t,s)\frac{\partial x(t,s)}{\partial s_k} + B(t,s)x(t,s)+f(t,s)+u(t,s), \]

where \(x=(x_1,\ldots,x_N)\), and the system is hyperbolic with smooth coefficients. The boundary conditions are prescribed in the form

\[ x(a,s)=x_0(s)\in W_2^{(1)}(R^n),\qquad \|x(b,s)-x_1(s)\|_{\mathscr L_2(R^n)}\leq \rho, \]

and the domain \(U\) is the ball:
\[ \{u(s):\|u(s)\|_{W_2^1(R^n)}\leq 1\}. \]

The existence of an optimal control can be proved by using the well-known inequality of I. G. Petrovskii \((3)\):

\[ \|x(t,s)\|_{\mathscr L_2(R^n)} \leq C\left\{ \|x_0(s)\|_{\mathscr L_2(R^n)} + \|f(t,s)+u(t,s)\|_{\mathscr L_2(R^n\times(a,b))} \right\}. \]

2)
\[ \frac{\partial^{2}x(t,s)}{\partial t^{2}}=\mathcal{L}x(t,s)+f(t,s)+u(t,s), \]

where \(\mathcal{L}x(t,s)\) is defined in (3). A mixed problem is considered in the cylinder \(Q=\Omega\times[a,b]\) with the conditions

\[ x(a,s)=x_0(s)\in W_2^{(2)}(\Omega),\qquad \frac{\partial x(a,s)}{\partial t}=x_1(s)\in W_2^{(1)}(\Omega),\qquad x(t,s)\big|_{\Gamma}=0. \]

Moreover, for \(t=b\) the inequality

\[ \left\|x(b,s)-x_2(s)\right\|_{W_2^{(1)}(\Omega)}^{2} + \left\|\frac{\partial x(b,s)}{\partial t}-x_3(s)\right\|_{\mathcal{L}_2(\Omega)}^{2} \leq \rho^{2}, \]

is satisfied, and the domain

\[ U=\{u(s):\|u(s)\|_{W_2^{(1)}(\Omega)}\leq 1\}. \]

3)
\[ \frac{\partial x(t,s)}{\partial t}=\mathcal{L}x(t,s)+f(t,s)+u(t,s), \]

where \(\mathcal{L}x(t,s)\) is defined in (3). The conditions on the boundary of \(Q\) have the form:

\[ x(a,s)=x_0(s)\in W_2^{(2)}(\Omega),\qquad x(t,s)\big|_{\Gamma}=0,\qquad \left\|x(b,s)-x_1(s)\right\|_{\mathcal{L}_2(\Omega)}\leq \rho, \]

and the domain

\[ U=\{u(s):\|u(s)\|_{W_2^{(1)}(\Omega)}\leq 1\}. \]

The existence of optimal controls in examples 2) and 3) is easy to prove by using the known a priori estimates.

Moscow State University
named after M. V. Lomonosov

Received
6 XII 1962

REFERENCES

1. L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze, E. F. Mishchenko, The Mathematical Theory of Optimal Processes, Moscow, 1961.
2. Yu. V. Egorov, DAN, 145, No. 4, 720 (1962).
3. I. G. Petrovskii, Matem. sbornik, 2 (44), 815 (1937).
4. S. L. Sobolev, Some Applications of Functional Analysis in Mathematical Physics, Leningrad, 1950.