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CYBERNETICS AND CONTROL THEORY
T. G. BABUNASHVILI
SYNTHESIS OF LINEAR OPTIMAL SYSTEMS*
(Presented by Academician L. S. Pontryagin on 25 XI 1963)
A number of works have been devoted to the synthesis of linear optimal systems; among them we note the work of Neustadt \((^1)\) and the work of N. N. Krasovskii \((^2)\). In \((^1)\) the synthesis problem for homogeneous linear systems is solved completely; in \((^2)\) the general case of nonhomogeneous linear systems is considered, but the method proposed there is too complicated.
Here we describe a new synthesis method, suitable for arbitrary nonhomogeneous (nondegenerate, see below) linear systems. In connection with the method presented here, see also the work of Antosievich \((^3)\).
1°. Formulation of the problem. Let the equation be given
\[ \dot{x}=A(t)x+B(t)u+f(t). \tag{1} \]
Here \(x\) is an \(n\)-dimensional phase column vector; \(A(t)\) is a summable \(n \times n\)-matrix (i.e., a matrix whose elements are summable on any bounded interval of the time axis); \(u\) is an \(r\)-dimensional control column vector; \(B(t)\) is a summable \(n \times r\)-matrix; \(f(t)\) is an \(n\)-dimensional summable column vector. The control \(u\) is sought in the class of measurable functions with values in a given convex compact polyhedron \(U\) of \(r\)-dimensional space, containing the origin.
The problem is as follows: for a given initial position \(x_0\) in the phase space, find an optimal control that transfers the phase point along the corresponding (optimal) trajectory of equation (1) from \(x_0\) to the origin in minimal time.
Write the equation
\[ \dot{\psi}=-\psi A(t), \tag{2} \]
where \(\psi\) is an \(n\)-dimensional row, and define the “norm” \(\|v\|\) in the space of \(r\)-dimensional row vectors \(v\) by the formula
\[ \|v\|=\max_{u \in U} vu. \]
The necessary condition for optimality (the maximum principle, see \((^4)\)) can now be formulated as follows.
For every optimal control \(u(t)\), \(0 \le t \le T\), there exists a nonzero solution \(\psi(t)\), \(0 \le t \le T\), of equation (2) such that almost everywhere on \(0 \le t \le T\)
\[ \psi(t)B(t)u(t)=\|\psi(t)B(t)\|. \tag{3} \]
Equation (1) is assumed to be nondegenerate (see \((^4)\)); this is equivalent to the assertion that, for any given nonzero solution \(\psi(t)\) of equation (2), the control \(u(t)\) is uniquely determined almost for all \(t\) from the maximum condition (3).
Thus, if from a given \(x_0\) one can reach the origin, then the optimal problem (for this given \(x_0\)) will be solved if we can
* The work was carried out in L. S. Pontryagin’s seminar on the theory of oscillations and automatic control.
compute the initial value \(\psi_{x_0}=\psi(0)\) of the corresponding solution \(\psi(t)\) of equation (2). The computation of the vector \(\psi_{x_0}\) from the vector \(x_0\) will be called the synthesis of the optimal system described by equation (1).
\(2^\circ\). Derivation of the basic equation (6) (see also \((2,3)\)). The solution \(x(t)\) of equation (1) with the initial condition \(x(0)=x_0\) has the form
\[ x(t)=\Phi(t)\left[x_0+\int_0^t \Phi^{-1}(\tau)\bigl(B(\tau)u(\tau)+f(\tau)\bigr)\,d\tau\right], \]
where \(\Phi(t)\) is the fundamental matrix for the homogeneous equation
\(\dot x=A(t)x\), normalized at \(t=0\). Let \(T_{x_0}\) be the optimal transition time from \(x_0\) to the origin. Finding the optimal control \(u_{x_0}(t)\), \(0\leq t\leq T_{x_0}\), is equivalent to solving the equation
\[ z(T)=-\left(x_0+\int_0^T \Phi^{-1}(t)f(t)\,dt\right) =\int_0^T \Phi^{-1}(t)B(t)u(t)\,dt =\int_0^T K(t)u(t)\,dt \tag{4} \]
with respect to the unknowns \(T\), \(u(t)\), \(0\leq t\leq T\), where \(T\) is taken to be the smallest positive root of this equation. The solution \(T_{x_0}\), \(u_{x_0}(t)\), \(0\leq t\leq T_{x_0}\), will be called the optimal solution of equation (4).
In order that, for any prescribed \(T>0\), equation (4) be solvable with respect to \(u(t)\in U\), \(0\leq t\leq T\), it is necessary and sufficient that, for an arbitrary \(n\)-dimensional row \(\chi\), the inequality
\[ \chi z(T)\leq \int_0^T \|\chi K(t)\|\,dt \tag{5} \]
hold.
The proof is given in \((3)\).
Theorem. In order that equation (4) have an optimal solution \(T_{x_0}\), \(u_{x_0}(t)\), \(0\leq t\leq T_{x_0}\), it is necessary and sufficient that there exist a nonzero row \(\psi_0\) satisfying the equation
\[ \psi_0 z(T_{x_0})= \int_0^{T_{x_0}} \|\psi_0 K(t)\|\,dt = \min_{\chi z(T_{x_0})=\psi_0 z(T_{x_0})} \int_0^{T_{x_0}} \|\chi K(t)\|\,dt, \tag{6} \]
(i.e. the minimum is taken over all \(\chi\) satisfying the condition
\(\chi z(T_{x_0})=\psi_0 z(T_{x_0})\)). Any solution \(\psi_0\) of equation (6) may be taken as the vector \(\psi_{x_0}\) and the optimal control \(u_{x_0}(t)\), \(0\leq t\leq T_{x_0}\), may be determined from the maximum condition (3), where \(\psi(t)\), \(0\leq t\leq T_{x_0}\), is the solution of equation (2) with initial condition \(\psi_{x_0}=\psi(0)\).
Proof. Let \(\chi\) be an arbitrary \(n\)-dimensional row satisfying the condition \(\chi z(T_{x_0})>0\), and let \(\alpha\chi z(T_{x_0})=\psi_0 z(T_{x_0})\); from the nondegeneracy of equation (1) it follows that the factor \(\alpha>0\). Therefore, from (6) there follows the inequality
\[ \alpha\chi z(T_{x_0})= \int_0^{T_{x_0}} \|\psi_0K(t)\|\,dt \leq \int_0^{T_{x_0}} \|\alpha\chi K(t)\|\,dt, \]
i.e. inequality (5), equivalent to equation (4). If, conversely, \(T_{x_0}\), \(u_{x_0}(t)\), \(0\leq t\leq T_{x_0}\), is an optimal solution of equation (4), then, according to the maximum principle, there exists a solution \(\psi(t)=\psi_{x_0}\Phi^{-1}(t)\) of equation (2) such that
\(\psi(t)B(t)u_{x_0}(t)=\|\psi_{x_0}K(t)\|\); consequently, multiplying (4) by \(\psi_{x_0}\), we obtain
\[ \psi_{x_0}z(T_{x_0}) = \int_0^{T_{x_0}} \psi_{x_0}K(t)u_{x_0}(t)\,dt = \int_0^{T_{x_0}} \|\psi_{x_0}K(t)\|\,dt, \]
i.e. equality (6).
It is easy to see that the control \(u_0(t)\), \(0 \leqslant t \leqslant T_{x_0}\), defined by the equation \(\psi_0 K(t)u_0(t)=\|\psi_0 K(t)\|\), where \(\psi_0\) is any nonzero solution of equation (6), is optimal: \(u_0(t)=u_{x_0}(t)\), \(0 \leqslant t \leqslant T_{x_0}\). Indeed, if \(u_0(t)\ne u_{x_0}(t)\) on a set of positive measure, then
\[ \psi_0 z(T_{x_0})=\int_0^{T_{x_0}}\psi_0K(t)u_{x_0}(t)\,dt < \int_0^{T_{x_0}}\psi_0K(t)u_0(t)\,dt = \int_0^{T_{x_0}}\|\psi_0K(t)\|\,dt, \]
which contradicts equality (6).
Thus, the synthesis problem is equivalent to solving equation (6) with respect to the unknowns \(\psi_0, T_{x_0}\), and as \(T_{x_0}\) one must take the least positive root of this equation.
\(3^\circ\). Solution of equation (6). Equation (6) can be solved by the method of gradient descent, based on the following proposition.
For any \(T>0\), the gradient of the function \(g(\chi)=\int_0^T\|\chi K(t)\|\,dt\) with respect to \(\chi\) is continuous; every relative minimum of the function \(g(\chi)\), under the condition \(\chi z(T)=\mathrm{const}>0\), is its absolute minimum (under the given condition \(\chi z(T)=\mathrm{const}\)).
Proof. In view of the nondegeneracy of equation (1),
\[ \|\chi K(t)\|=\chi K(t)v_\chi(t), \]
where \(v_\chi(t)\) is a piecewise constant function, depending on \(\chi\ne0\), on a set of full measure (in \(t\)), with values at the vertices of the polyhedron \(U\); under a small change of \(\chi\), the function \(v_\chi(t)\) changes on a set of small measure. Consequently,
\[ \operatorname{grad} g(\chi)=\int_0^T K(t)v_\chi(t)\,dt \]
changes continuously together with \(\chi\). Let \(\chi_1,\chi_2\) be two stationary points of the function \(g(\chi)\) under the condition \(\chi z(T)=\mathrm{const}>0\); we shall show that \(g(\chi_1)=g(\chi_2)\). Suppose the contrary, and let \(g(\chi_1)>g(\chi_2)\). We have
\[ \operatorname{grad} g(\chi_i)=\int_0^T K(t)v_{\chi_i}(t)\,dt = \lambda_i z(T), \]
\[ \chi_i\cdot \operatorname{grad} g(\chi_i) = \int_0^T \chi_iK(t)v_{\chi_i}(t)\,dt = \int_0^T \|\chi_iK(t)\|\,dt = g(\chi_i) = \lambda_i\cdot\mathrm{const},\quad i=1,2; \]
therefore \(\lambda_1>\lambda_2\).
Next we have:
\[ \int_0^T K(t)\bigl(v_{\chi_1}(t)-v_{\chi_2}(t)\bigr)\,dt = (\lambda_1-\lambda_2)z(T); \]
multiplying both sides by \(\chi_2\), we obtain the relation
\[ \int_0^T \chi_2K(t)v_{\chi_1}(t)\,dt - \int_0^T \|\chi_2K(t)\|\,dt = (\lambda_1-\lambda_2)\cdot\mathrm{const}>0, \]
which is contradictory, since
\[ \int_0^T \chi_2K(t)v_{\chi_1}(t)\,dt \leqslant \int_0^T \|\chi_2K(t)\|\,dt. \]
The proposition just proved gives the following method for solving equation (6). Choose a “first approximation” \(\chi_1\) to the solution \(\psi_0\), subjecting it to the sole condition \(\chi_1 z(0)>0\), and begin increasing the time \(t\) from 0 to the first instant \(t_1\) (the “first approximation” to \(T_{x_0}\)) when
\[ \chi_1 z(t_1)=\int_0^{t_1}\|\chi_1K(t)\|\,dt \]
(if for every \(t>0\)
\[ \chi_1 z(t)>\int_0^t\|\chi_1K(\tau)\|\,d\tau, \]
then the optimal problem with the given initial value \(x_0\), obviously, has no solution). After this, by the method of gradient descent we find the minimum of the function \(g_1(\chi)=\)
\[ = \int_0^{t_1} \|\chi K(t)\|\,dt \]
under the condition \(\chi z(t_1)=\chi_1 z(t_1)\). If the minimum point \(\chi_2 \ne \chi_1\), then
\[ \chi_2 z(t_1) > \int_0^{t_1} \|\chi_2 K(t)\|\,dt, \]
and we shall begin to increase the time from the instant \(t_1\) to the instant \(t_2\), when again
\[ \chi_2 z(t_2) = \int_0^{t_2} \|\chi_2 K(t)\|\,dt; \]
we obtain the “second approximations” \(t_2, \chi_2\), and so on. It is easy to see that the increasing sequence \(t_1 \leq t_2 \leq \cdots\) has a finite least upper bound if and only if the optimal problem with the given initial value \(x_0\) has a solution, and this least upper bound is equal to the optimal time \(T_{x_0}\). In the case of finite \(T_{x_0}\), the sequence of unit vectors
\[ \frac{\chi_1}{\|\chi_1\|},\quad \frac{\chi_2}{\|\chi_2\|}, \ldots \]
(\(\|\chi\|\) is an arbitrary vector norm) converges to some compact set of vectors that constitute all unit-length solutions of equation (6).
Mathematical Institute named after V. A. Steklov
Academy of Sciences of the USSR
Received
24 X 1963
REFERENCES
- L. W. Neustadt, J. Math. Anal. and Appl., 1, No. 4, 484 (1960).
- N. N. Krasovskii, Matem. sborn., 53 (95), 2, 195 (1961).
- H. A. Antosiewicz, Arch. Rat. Mech. and Anal., 12, 4, 313 (1963).
- L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze, E. F. Mishchenko, Mathematical Theory of Optimal Processes, Moscow, 1961.