ON STABILIZATION OF DYNAMIC SYSTEMS BY SUPPLEMENTARY FORCES1

N. N. KRASOVSKII

1. In the theory of controlled systems, and in particular in the theory of optimal processes, which has developed in recent years, problems of stabilizing (of optimal stabilization of) motions of dynamic systems by supplementary actions play an essential role. These problems have been solved by methods of the theory of optimal processes based on the maximum principle [1], by methods of dynamic programming theory [2], by methods of the classical calculus of variations, in the theory of the analytic construction of regulators [3], etc. A large number of works of an applied character have been devoted to stabilization problems.

The aim of the present report is to discuss certain general results of the mathematical theory of stabilization of dynamic systems. Here the main attention is given to two circumstances: 1) stabilization problems are considered as a natural development of Lyapunov’s general problem on stability of motion [4], 2) stabilization problems are considered in connection with problems of control and observation in dynamic systems. The first circumstance makes it possible to bring to the study of stabilization problems the methods and results developed and accumulated in the theory of stability of motion. In this connection, the works of N. G. Chetaev [5] and his school, devoted to the development of Lyapunov’s function method, are used essentially. An important place in the theory of stabilization of dynamic systems is occupied by qualitative methods of the theory of stability, the development of which was begun by N. P. Erugin [6] and his school. The connection of problems of stabilization of motions with the properties of controllability and observability of dynamic systems became clear in the course of the development of the theory of optimal processes, and in particular in the course of the development of A. M. Letov’s problem [3]. Problems of control and observation in linear systems under the condition of optimization of quadratic quality estimates were studied by R. Kalman [7], and a principle of duality between control and observation was formulated. In the report, linear problems of control and observation are treated [8] as problems of determining a linear operation satisfying certain conditions on prescribed elements of a suitable functional space. Such a point of view, proposed earlier in [9], makes it possible to bring to the study of problems of control and observation general methods of functional analysis, above all the theory of the moment problem [10] and the theory of problems of separation of sets [11]. The latter circumstance makes it possible to note the connection of the problems under consideration with problems of game theory [12].

2.

Let us dwell briefly on some results of the theory of control and observation in linear dynamical systems. Suppose that the controlled system is described by the vector differential equation

\[ \frac{dx}{dt}=Ax+Bu+f(t), \tag{2.1} \]

where \(x\) is an \(n\)-vector of phase coordinates; \(u\) is an \(r\)-vector of control forces; \(f(t)\) is an \(n\)-vector of external actions; \(A,\ B\) are continuous matrices of the corresponding dimensions. In its simplest form, the control problem is formulated as follows.

Problem 2.1. Given the initial and final states of the system \(x^0,\ x^{(k)}\) and the time interval \(0 \le t \le T\). It is required to find a control \(u(t)\) \((0 \le t \le T)\) that transfers system (2.1) from the state \(x(0)=x^0\) to the state \(x(T)=x^{(k)}\).

In addition, there may be given an additional condition requiring the minimum of some specified intensity \(\rho^*[u]\) of the signal \(u(t)\) (for example, \(\rho^*[u]=\max(|u(t)|,\ 0\le t\le T)=\min\)).

The following result is valid [8]. Let \(h_i[t,\tau;j]\) \((i=1,\ldots,n;\ j=1,\ldots,r)\) be the impulse transition function of system (2.1) corresponding to the \(i\)-th coordinate \(x\) and the \(j\)-th coordinate \(u\). In other words, \(h_i[t,\tau;j]=x_i(t)\) on the motion of system (2.1) generated by the conditions \(x(0)=0,\ u_k(t)\equiv 0\ (k\ne j),\ u_j(t)=\delta(t-\tau)\), where \(\delta\) is the delta function. By the symbol \(g_i[t,\tau;k]\) \((i,k=1,\ldots,n)\) we shall denote the impulse transition function of system (2.1) corresponding to the \(i\)-th coordinate \(x\) and the \(k\)-th coordinate \(f\). Without loss of generality, in the conditions of Problem 2.1 we take \(x^0=0,\ x^{(k)}=q\). We shall also assume that the minimized intensity \(\rho^*[u]\) of the action \(u(t)\) can be interpreted as the norm of a linear functional [11]

\[ U[\xi]=\int_0^T\left(\sum_{j=1}^r \xi_j(\tau)u_j(\tau)\right)d\tau, \tag{2.2} \]

defined on some space of vector functions \(\xi(\tau)=\{\xi_j(\tau)\}\) \((j=1,\ldots,r;\ 0\le \tau\le T)\). We shall denote the norm of the space \(\{\xi\}\) by the symbol \(\rho(\xi)\). This assumption covers a sufficiently large number of cases of applied interest. Then Problem 2.1 has a solution \(u(t)\) if and only if

\[ \rho(\xi^0)=\mu>0. \tag{2.3} \]

Here \(\xi^0(\tau)\) is the solution of the variational problem:

\[ \rho(\xi[\tau,l])=\min \quad \text{for} \quad \sum_{i=1}^n l_i c_i=1, \tag{2.4} \]

\[ c_i=q_i-\int_0^T\left(\sum_{k=1}^n g_i[T,\tau;k]f_k(\tau)\right)d\tau, \tag{2.5} \]

\[ \xi_j[\tau,l]=\sum_{i=1}^n h_i[T,\tau;j]l_i\ (l_i=\mathrm{const}). \tag{2.6} \]

The optimal control action \(u^0(t)\) with minimal intensity \(\rho^*\) satisfies the conditions:

\[ \rho^*[u^0]=\frac{1}{\mu}, \tag{2.7} \]

\[ U^0[\xi^0]=\int_0^T \left(\sum_{j=1}^n \xi_j^0(\tau)u^0(\tau)\right)d\tau =\max U[\xi^0]= \]

\[ =\max \left[\int_0^T \left(\sum_{j=1}^n \xi_j^0(\tau)u(\tau)\right)d\tau\right] \quad \text{for } \rho^*[u]=\frac{1}{\mu}. \tag{2.8} \]

This result can be interpreted quite naturally from the point of view of game theory, if one uses the duality principle [7] between control and observation.

The linear problem of observation can be formulated as follows.

Problem 2.2. Find a linear operation \(\varphi\) that reconstructs the quantity

\[ z(T)=e_1x_1(T)+\ldots+e_nx_n(T) \tag{2.9} \]

from observation of the quantity

\[ y(t)=Px(t) \tag{2.10} \]

on the interval \(0\le t\le T\), where \(x(t)\) is the motion of the system

\[ \frac{dx}{dt}=Dx. \tag{2.11} \]

In (2.9)—(2.11) we shall regard \(x\) and \(e\) as \(n\)-vectors, and \(y\) as an \(r\)-vector; the operation \(\varphi\) is then sought in the form

\[ z(T)=\varphi[y]=\int_0^T \left(\sum_{j=1}^r y_j(\tau)\omega_j(\tau)\right)d\tau, \tag{2.12} \]

where \(\omega(\tau)\) is the vector-function sought.

If the quantity \(y(t)\) is fed into the operation (2.12) with an error \(s(t)\), i.e., if in (2.12), instead of \(y_j(\tau)\), we have the quantities \(y_j(\tau)+s_j(\tau)\), and if an estimate \(\rho(s)\le \Delta\) is given for the intensity \(\rho(s)\) of the error \(s(t)\) on the interval \(0\le t\le T\), then Problem 2.2 is naturally accompanied by the condition

\[ \rho^*(\omega)=\min, \tag{2.13} \]

since the error of the operation (2.12) is equal to \(\Delta \rho^*(\omega)\).

The duality rule between Problems 2.1 and 2.2 is as follows: under the conditions \(A=D^*\), \(B=P^*\), \(e=c\), Problems 2.1 and 2.2 are equivalent, i.e., the solution \(u^0(t)\) of Problem 2.1 coincides with the solution \(\omega^0(t)\) of Problem 2.2. The quantity \(\xi^0(\tau)\), which determines by (2.3)—(2.8) the solution of Problem 2.1 on control in terms of the dual Problem 2.2 on observation, receives the following interpretation. The vector-function \(\xi^0(\tau)\) (2.3) coincides with the observed signal \(y^0(\tau)\) (2.10) from the dual observation problem, having the least possible intensity \(\rho(y)\) under the condition \(z(T)=1\). We shall say that the signal \(y(t)\) \((0\le t\le T)\) ne-

sets the value \(z(T)=1\), if \(y(t)\) (2.10) corresponds to the motion \(x(t)\) of system (2.11), for which in (2.9) \(z(T)=1\); if, moreover, \(\rho(y^0)=\min\), then the signal \(y^0(t)\) will be called minimal. Now the results concerning the solution of Problems 2.1 and 2.2 are formulated as follows.

Theorem 2.1. Problem 2.2 has a solution \(w(t)\) if and only if there exists a minimal signal \(y^0(t)\), different from the identically zero one, carrying the value \(z(T)=1\). Let \(\rho(y^0)=\mu>0\). Then the optimal signal \(w^0(t)\) solving Problem 2.2 has intensity \(\rho^*[w^0]=1/\mu\) and is distinguished among all other signals \(w(t)\) with \(\rho^*[w]=1/\mu\) by the condition that, for \(w=w^0\), the operation

\[ \varphi[y^0]=\int_0^T \left(\sum_{j=1}^{r} y_j^0(\tau) w_j(\tau)\right)\,d\tau \tag{2.14} \]

gives the greatest possible result.

Problem 2.1 has a solution \(u(t)\) if and only if there exists a minimal signal \(y^0(t)\), different from the identically zero one, carrying the value \(z(T)=1\) in the dual observation problem 2.2. The optimal control \(u^0(t)\) solving Problem 2.1 has intensity \(\rho^*[u^0]=1/\mu\), where \(\mu=\rho(y^0)\), and is distinguished among all other signals \(u(t)=w(t)\) with \(\rho^*[u]=1/\mu\) by the fact that, for \(w=u^0(t)\), operation (2.14) gives the greatest result. Observation Problem 2.2 may be associated with the game:

Game 2.1. The strategies of the first player are signals \(w(t)\) with intensity \(\rho^*[w]=1\); the strategies of the second player are signals \(y(t)\) (2.10) carrying the value \(z(T)=1\). The payoff function is

\[ \varphi=\int_0^T \left(\sum_{j=1}^{r} y_j(\tau) w_j(\tau)\right)\,d\tau . \]

The aim of the first player is the greatest possible value of \(\varphi\), and the aim of the second player is the smallest possible value of \(\varphi\).

Game 2.1 has a saddle point \(\{y^0(t), w^{0*}(t)\}\), where \(y^0(t)\) is the minimal signal, and the signal \(w^{0*}(t)\) differs only by the factor \(1/\mu\) from the solution \(w^0(t)\) of Problem 2.2 having the least intensity \(\rho^*(w)\). The signal \(w^{0*}(t)\) determines that functional \(\varphi\) whose surface \(\varphi=\mathrm{const}\) separates [11] the linear set \(\{y(\tau)\}\) of the second player’s strategies.

Thus, the optimal solution \(w^0(t)\) of Problem 2.2 is interpreted as that operation which gives the greatest possible value in the most unfavorable case \(y^0(t)\) of the observed signal \(y(t)\). From this point of view, the conditions of the maximum principle [1] are interpreted here as the conditions \(\max_w \varphi\) in Game 2.1; here it is clarified that the vector \(\Psi(t)\) which appears in the maximum principle and in this case enters the expression for \(y(t)\) must satisfy the condition \(\min_y \varphi\).

3. Let us consider the problem of stabilization. In its simplest form, the linear problem of stabilization is formulated as follows.

Problem 3.1. The motion of the controlled system is described by the equation

\[ \frac{dx}{dt}=Ax+Bu. \tag{3.1} \]

Find a function

\[ u=Gx \tag{3.2} \]

such that the motion \(x=0\) of system (3.1), (3.2) is asymptotically stable.

The connection of Problem 3.1 with control problems is determined by the following circumstance. The phase space \(\{x\}\) can be regarded as the direct sum of two subspaces \(N^{(1)}\) and \(N^{(2)}\). Initial conditions \(x^{(1)}\) from \(N^{(1)}\) generate motions \(x^{(1)}(t)\) of the system

\[ \frac{dx}{dt}=Ax, \tag{3.3} \]

which do not converge to the point \(x=0\) as \(t\to\infty\); initial conditions \(x^{(2)}\) from \(N^{(2)}\) generate motions \(x^{(2)}(t)\) of (3.3) which asymptotically approach the point \(x=0\) as \(t\to\infty\). In the case of a simple structure of the constant matrix \(A\), one may choose as \(N^{(1)}\) the subspace \(N^{+,0}\) generated by the eigenvectors \(l\) of the matrix \(A\) corresponding to eigenvalues \(\lambda\) with nonnegative real part; as \(N^{(2)}\) in this case one should choose the subspace \(N^{-}\), generated by the eigenvectors \(l\) corresponding to characteristic roots \(\lambda\) with \(\operatorname{Re}\lambda<0\). For Problem 3.1 to be solvable it is necessary and sufficient that, for system (3.1), Problem 2.1 on control be solvable for all points \(q\) from the subspace \(N^{(1)}\). From this one derives effective conditions for stabilizability of the linear system (3.1). Suppose, for example, that the matrices \(A\) and \(B\) are constant. The solvability conditions for Problem 2.1, given in Section 2, may be formulated as follows: in order that, for system (3.1), Problem 2.1 have a solution with \(x(0)=0,\ x(T)=q\ (T>0)\), it is necessary and sufficient that the point \(q\) belong to the subspace \(N^{(s)}\) generated by the column vectors of the matrix

\[ S=(A,\ AB,\ \ldots,\ A^{n-1}B). \tag{3.4} \]

Hence follows the stabilizability criterion [13, 14]. Denote by \(Q\) the matrix that reduces \(A\) to Jordan form \(G=Q^{-1}AQ\). Suppose that on the main diagonal of \(G\), in the \(i\)-th column, there stands the number \(\lambda=\lambda_i^{*}\); then we shall say that the \(i\)-th column \(Q_i\) of the matrix \(Q\) corresponds to the eigenvalue \(\lambda_i^{*}\) of the matrix \(A\). In the case of a simple structure of \(A\), the vectors \(Q_i\) coincide with the corresponding eigenvectors of \(A\). Denote by \(Q_i^{+}, Q_j^{0}, Q_k^{-}\) the columns of \(Q\) corresponding to eigenvalues \(\lambda\) of the matrix \(A\) with \(\operatorname{Re}\lambda>0,\ \operatorname{Re}\lambda=0\), and \(\operatorname{Re}\lambda<0\), respectively. Then for Problem 3.1 to be solvable it is necessary and sufficient that all vectors \(Q_i^{+}\) belong to the subspace \(N^{(s)}\), generated by the vectors (3.4), which we shall denote symbolically by

\[ Q^{+}\subset N^{(s)}. \tag{3.5} \]

Problem 3.1 has more than one solution. A concrete computation of the matrix \(G\) in (3.2) can be carried out by solving, for (3.1), the problem of the analytic construction of an optimal regulator [3], minimizing the quantity

\[ I=\int_{0}^{\infty}\left\{\omega[x]+\sum_{j=1}^{r}u_j^2(t)\right\}dt=\min, \tag{3.6} \]

where \(\omega[x,t]\) is a nonnegative quadratic form. To solve problem (3.6) one must find the optimal Lyapunov function \(v^0(x)\), satisfying the Lyapunov–Bellman equation [2, 3, 13],

\[ \min_u \left[\frac{dv^0}{dt}+\omega[x]+\sum_{j=1}^{r}u_j^2\right] = \frac{dv^0}{dt}+\omega[x]+\sum_{j=1}^{r}\bigl(u_j^0[x]\bigr)^2=0, \tag{3.7} \]

where \(u^0[x]\) is the solution of problem 3.1. The function \(v^0\) is a quadratic form. If \(\omega[x]\) is a positive definite quadratic form of the variables \(x_i\) \((i=1,\ldots,n)\), then from the positive definiteness of the function \(v^0\) satisfying equation (3.7), by virtue of Lyapunov’s theorem on asymptotic stability [4], the asymptotic stability of the motion \(x=0\) of the system immediately follows:

\[ \frac{dx}{dt}=Ax+Bu^0[x], \tag{3.8} \]

where \(u^0[x]\) is the solution of equations (3.7). Indeed, in this case the function \(v^0\) has, by virtue of equations (3.8), a negative definite derivative

\[ \frac{dv^0}{dt}=-\omega[x]+\sum_{j=1}^{r}u_j^0[x]. \tag{3.9} \]

For reasons of computational convenience, or for other reasons, it is sometimes expedient to choose the function \(\omega[x]\) only sign-positive. Then the derivative \(dv^0/dt\) (3.9) may turn out to be only sign-negative, and additional considerations are needed to verify asymptotic stability. We note here one such consideration, connecting the stabilization problem with the observability property of a dynamical system considered in Section 2. The following stability criterion is valid.

Let \(\omega[x]\) be a sign-positive quadratic form. Denote by \(N^{(\omega)}\) the subspace of the space \(\{x\}\) where \(\omega[x]=0\).

Lemma 3.1. Consider the stationary system (3.8), where \(u^0[x]=Gx\). If one can specify a positive definite function \(v^0[x]\) having a sign-negative derivative (3.9), and if, for system (3.3), problem 2.2 on observation of the quantity \(z(T)=px\) from the quantity \(y(t)=u^0(t)=Gx(t)\) is solvable for all vectors \(p\) from \(N^{(\omega)}\), then the motion \(x=0\) of system (3.8) is asymptotically stable.

Indeed, consider a motion \(x(t)\) of system (3.8) on which \(dv^0/dt\equiv0\). This is possible only under the conditions \(u^0[x(t)]\equiv0\) and \(\omega[x(t)]\equiv0\), i.e., under the condition that the motion \(x(t)\) lies in the subspace \(N^{(\omega)}\) for all \(t>0\). If \(x(t)\not\equiv0\), then at some instant \(t=T\) we have \(x(T)=p\ne0\) and \(p\in N^{(\omega)}\), with \(z(T)=x(T)p=\alpha>0\). By the condition of the lemma, the quantity \(z(T)\) is observable from the quantity \(u^0[x(t)]\) \((0\le t\le T)\), and this is possible, according to Section 2, only under the condition that \(u^0[x(t)]\not\equiv0\), if \(z(T)\ne0\). The contradiction obtained \(\bigl(u^0[x(t)]\equiv0,\ u^0[x(t)]\not\equiv0\bigr)\) shows that there are no motions \(x(t)\) different from \(x(t)\equiv0\) for which \(dv^0/dt\equiv0\). According to the theorem from [15], asymptotic stability of the motion \(x=0\) of system (3.8) follows from this. The lemma is proved.

The quantity \(z=px\) is observable from the quantity \(y=Gx\) if [7, 13] the vector \(p\) is contained in the subspace \(N^{(R)}\) of row vectors of the matrix

\[ R=(G,\ GA,\ \ldots,\ GA^{n-1}). \tag{3.10} \]

Consequently, for asymptotic stability of system (3.8), which has a positive definite Lyapunov function \(v^0\) with sign-negative derivative \(dv^0/dt\) (3.9), it is sufficient that the linear

subspace \(N^{(\omega)}\), where \(\omega[x]=0\), was contained in the subspace \(N^{(R)}\), generated by the row vectors of the matrix (3.10).

In conclusion of this section, let us note that the condition for stabilization of a dynamical system considered here, consisting in the controllability of the intrinsically unstable degrees of freedom of the system, is not a particular circumstance inherent only in a system of the form (3.1), but has a fairly general character, as will be shown below by the example of systems with aftereffect.

1. Let us consider the problem of stabilization of a nonlinear system. Suppose the perturbed motions of a controllable system are described by the equation

\[ \frac{dx}{dt}=Ax+Bu+f[x,u,t], \tag{4.1} \]

where the vector function \(f\) has, in a neighborhood of the point \(x=0,\ u=0\), order of smallness higher than the first.

Problem 4.1. Find a function

\[ u=g[x,t] \tag{4.2} \]

such that the motion \(x=0\) of the system (4.1), (4.2) is asymptotically stable in the sense of Lyapunov.

Investigation of Problem 4.1 in the linear approximation by the methods described in Section 3 makes it possible to formulate stabilization criteria which naturally generalize the theory of Lyapunov stability of motion by the first approximation to the case of controllable systems. The following assertion is valid [14]. (We restrict ourselves to the case \(A=\mathrm{const}\), \(B=\mathrm{const}\).)

Theorem 4.1. If all the vectors \(Q_i^{+}, Q_j^{0}\) belong to the space \(N^{(s)}\), generated by the columns of the matrix (3.4), then Problem 4.1 is solvable by a linear control \(u=Gx\), independently of the terms \(f\) of higher order of smallness. If at least one vector \(Q_i^{+}\) does not lie in \(N^{(s)}\), then Problem 4.1 is not solvable independently of the terms \(f\) of higher order of smallness. If all the vectors \(Q_i^{+}\) lie in \(N^{(s)}\), but at least one vector \(Q_j^{0}\) does not lie in \(N^{(s)}\), then there is a critical case of stabilization, when the solvability of Problem 4.1 and the form of the stabilizing action \(u[x,t]\) are determined by the terms \(f\) of higher order of smallness.

It should be emphasized that the investigation of the solvability of the stabilization problem in critical cases and the computation of the control \(u[x,t]\) are carried out by methods which are the result of reducing Problem 4.1 to the known and, by the present time, very fully developed theory of critical cases of Lyapunov stability [1, 2, 16, 17]. At the same time, however, certain features also arise, connected with the circumstance that the possibilities of stabilization in critical cases can, generally speaking, be extended by considering nonanalytic controls \(u\) containing terms depending on quantities of the form \(|\alpha_1 x_1+\ldots+\alpha_n x_n|\). This requires some additional development of the theory of critical cases.

1. Above, certain problems of control, observation, and stabilization were considered for systems with a finite number of degrees of freedom. The general theory, based on the approach to problems described in Sections 2–4, is developed without principal difficulties also for systems with an infinite number of degrees of freedom, since the apparatus of functional analysis (the moment problem, the problem of separation of sets, etc.), on which the results of Section 2 are based,

and the apparatus of stability theory, on which the results of Sections 3 and 4 are based in their connection with the results of Section 2, also work in the case of infinite-dimensional systems. However, an effective verification of the conditions of controllability, observability, and stabilizability of infinite-dimensional systems in cases of applied interest, and in particular in cases of interest for mechanics, apparently presents serious difficulties. Likewise, in these cases the concrete computation of controls and resolving signals also presents serious difficulties. These difficulties, as a rule, become still greater if the motion of an infinite-dimensional system is described in the functional space \(\{x\}\) by the equation \(d x/dt = Ax + Bu\) with an unbounded operator.

It is possible, however, to indicate classes of infinite-dimensional systems for which one can construct a sufficiently effective theory of stabilization, as well as investigate the problem, essential for systems with an infinite number of degrees of freedom, of approximating them by finite-dimensional systems. Examples of such systems may be systems with aftereffect, described by differential equations with time delays. The present section will be devoted to some results in the theory of optimal stabilization of linear systems with delays. The basis for these results is provided by the consideration of motions of systems with aftereffect in a suitable functional space [18] and the theory of Lyapunov’s function method developed on this basis for such systems [18], the theory of canonical decomposition of motions [19, 20], and the theory of the analytic construction of optimal regulators [21].

Consider a controlled system with aftereffect, described by the differential vector equation with time delay,

\[ \frac{dx}{dt}=Ax(t)+A_{\tau}x(t-\tau)+Bu, \tag{5.1} \]

where \(x\) is an \(n\)-vector of phase coordinates; \(u\) is an \(r\)-vector of control forces; \(\tau>0\) is a constant delay of the actions; \(A\), \(A_{\tau}\), \(B\) are constant matrices. For a specified control \(u(t)\), for times \(t \le t_0\) the motion \(x(t)\) of system (1.1) is determined by the prehistory \(x(t_0+\vartheta)=z(\vartheta)\) \((-\tau \le \vartheta \le 0)\) of the process. Therefore, if the problem is posed of stabilizing the motions of system (5.1) by feedback control, i.e., the problem of constructing the control \(u\) as a function of the current state of the system, then the quantity \(u\) in equation (5.1) should be sought in the form of a functional

\[ u(t)=\xi[x(t+\vartheta)]\quad (-\tau \le \vartheta \le 0). \tag{5.2} \]

The quality of the transient process \(x(t_0,t,z(\vartheta),u)\), generated in system (5.1) by the initial perturbation \(x(t_0+\vartheta)=z(\vartheta)\) \((-\tau \le \vartheta \le 0)\), for the chosen control \(u\), will be estimated by the quantity

\[ I[t_0,z(\vartheta),u]= \int_{t_0}^{\infty} \left( \omega[x(t,t_0,z(\vartheta),u)]+\sum_{j=1}^{r}u_j^2(t) \right)\,dt, \tag{5.3} \]

where \(\omega\) is a prescribed positive-definite form in \(x_i\). We discuss the following problem of analytic construction of an optimal regulator [3, 21], which includes the problem of stabilization of the motion of system (5.1).

Problem 5.1. Find a control

\[ u(t)=\xi^0[x(t+\vartheta)], \tag{5.4} \]

under which the motion \(x=0\) of system (5.1), (5.4) is asymptotically stable [18] and for which the inequality

\[ I[t_0,z(\vartheta),\xi^0]\leq I[t_0,z(\vartheta),\xi], \tag{5.5} \]

holds, whatever the initial conditions \(t_0,\ z(\vartheta)\) and the control law \(u=\xi[x,(t+\vartheta)]\) (5.2) for which there exist solutions \(x(t)\) of equation (5.1) and the quantity \(I[t_0,z(\vartheta),\xi]\) is meaningful.

In order to formulate a criterion for solvability of Problem 5.1 (in the case \(r=1\) this criterion was given in [22]), consider the linear equation with time lead, which may be regarded as the equation adjoint to equation (5.1) (for \(u\equiv0\)):

\[ \frac{dx}{dt}=-A^*x(t)-A_\tau^*x(t+\tau). \tag{5.6} \]

Here \(*\) denotes transposition. By an eigenvector \(l\) of system (5.6), corresponding to an eigenvalue \(\rho\), we shall mean a nonzero vector \(l\) satisfying equation (5.6) for \(x(t)=l\exp \rho t\).

Let \(\lambda_1,\ldots,\lambda_k\) be all the eigenvalues of the equation

\[ \frac{dx}{dt}=Ax(t)+A_\tau x(t-\tau), \tag{5.7} \]

satisfying the condition

\[ \operatorname{Re}\lambda_i\geq0 \tag{5.8} \]

(by eigenvalues \(\lambda_j\) of equation (5.7) we mean numbers for which equation (5.7) has solutions of the form \(x(t)=s\exp\lambda_j t\) with \(s\ne0\)). It is known [23] that the numbers \(\lambda_j\) are roots of the equation

\[ \left|A+A_\tau\exp(-\tau\lambda)-\lambda E\right|_1^n=0, \tag{5.9} \]

where \(E\) is the identity matrix. Let \(\rho_i=-\lambda_i\) \((i=1,\ldots,k)\) be the eigenvalues of equation (5.6) corresponding to the eigenvalues \(\lambda_i\) (5.8). Suppose that to the number \(\rho_i\) there correspond \(k_i\) linearly independent eigenvectors \(l[i,j]\) \((j=1,\ldots,k_i)\) of equation (5.6). The following assertion is valid:

Problem 5.1 is solvable if and only if the rank of each matrix \(H^{(i)}\)

\[ H^{(i)}=\{a_{jm}^{(i)}=l[i,j]b[m]\}\quad (i=1,\ldots,k;\ j=1,\ldots,k_i;\ m=1,\ldots,r) \]

is equal to \(k_i\). Here \(b[m]\) \((m=1,\ldots,r)\) are the columns of the matrix \(B\).

The condition formulated means controllability by the action \(u\) of all intrinsically unstable degrees of freedom of system (5.1) with aftereffect. These canonical degrees of freedom are singled out in the expansion [19, 20] of the motion \(x(t)\) (5.1) in eigen-elements of the form \(s\exp\lambda_i t\).

The optimal stabilizing action \(\xi^0[x(\vartheta)]\) has the form of a linear functional [21] of \(x(\vartheta)\)

\[ \xi_j^0[x(\vartheta)] = \sum_{i=1}^{n} \left( \nu_i^{(j)}x_i(0) + \int_{-\tau}^{0}\mu_i^{(j)}(\vartheta)x_i(\vartheta)\,d\vartheta \right) \quad (j=1,\ldots,r), \tag{5.10} \]

and, consequently, the optimal control law \(u(t)\) in system (5.1) has the form

\[ u(t)=\xi^0[x(t+\vartheta)] =\sum_{i=1}^{n}\left(\nu_i x_i(t)+\int_{-\tau}^{0}\mu_i(\vartheta)x_i(t+\vartheta)\,d\vartheta\right), \tag{5.11} \]

where \(\nu_i\) are constant \(j\)-vectors; \(\mu_i(\vartheta)\) are continuous \(j\)-vector functions.

The computation of the quantities \(\nu_i^{(j)}\) and \(\mu_i^{(j)}\) for (5.10) is difficult (it leads to partial differential equations). Therefore it is expedient to approximate the problem by an analogous problem for some suitable system with a finite number of degrees of freedom, described by ordinary differential equations. Such a suitable finite-dimensional system may be constructed either on the basis of the canonical expansion of the motions \(x(t)\) of (5.1) in terms of the proper elements of the free system (5.7), or by replacing the delay element \(y(t)=x(t-\tau)\) by a sequence of differentiating elements, proceeding from Taylor’s formula \(y(t+\tau)=x(t)+(dx/dt)\tau+O(\tau)\), etc. One such approximation was considered in [24], but without investigating the convergence of the approximations. The solution of Problem 5.1 on the basis of approximating the delay element by a sequence of differentiating elements was developed and tested at the computing center of Ural University by Yu. M. Repin and V. E. Tretyakov. We shall discuss here the justification of such an approximation.

When the delay element is replaced by a sequence of \(m\) differentiating elements, one arrives at the following system of ordinary differential equations approximating system (5.1):

\[ \frac{dy^{(i)}}{dt}=-my^{(i)}+my^{(i-1)}\quad (i=1,\ldots,m), \]

\[ \frac{dy^0}{dt}=Ay^0+A_\tau y^{(m)}+Bu, \tag{5.12} \]

where \(m\) is a natural number; \(y^0, y^{(i)}\) are \(n\)-dimensional vectors approximating the quantities \(x(t), x(t-i/m)\); \(A, A_\tau, B\) are the matrices from (5.1). (Without loss of generality, we put \(\tau=1\).) The vector \(y^0, y^{(i)}\) \((i=1,\ldots,m)\) will be denoted by the symbol \(\{y\}_m\). In accordance with (5.3), we shall estimate the quality of the transient process \(\{y(t)\}_m\) in system (5.12) by the quantity

\[ I_m[t_0,\{z\}_m,u] =\int_{t_0}^{\infty} \left(\omega[y^0(t,t_0,\{z\}_m,u)]+\sum_{j=1}^{r}u_j^2(t)\right)\,dt, \tag{5.13} \]

where \(y^0(t,t_0,\{z\}_m,u)\) is the component of the motion \(\{y(t)\}_m\) of system (5.12), generated by the initial condition \(\{y(t_0)\}_m=\{z\}_m\) under the control \(u\).

The auxiliary problem approximating Problem 5.1 is as follows.

Problem \(5.1^0\). Find a control

\[ u(t)=\eta^0[\{y(t)\}_m], \tag{5.14} \]

for which the motion \(y=0\) of system (5.12), (5.14) is asymptotically stable and for which the inequality

\[ I_m[t_0,\{z\}_m,\eta^0]\leq I_m[t_0,\{z\}_m,\eta] \tag{5.15} \]

holds.

whatever the initial conditions \(t_0, \{z\}\) and the control law
\(u=\eta[t,\{y(t)\}_m]\) for which solutions \(\{y(t)\}_m\) of equations (5.12) exist and the quantity \(I_m[t_0,\{z\},\eta]\) is meaningful.

Auxiliary problem 5.1 has a solution in the form of a linear function [3, 7] of \(y_i^0, y_i^{(j)}\)

\[ \eta_i^0[\{y\}_m] = \sum_{i=1}^{n} \left( \nu_i^{(j)}[m]y_i^0 + \sum_{k=1}^{m}\mu_i^{(j)}[m,k]y_i^{(k)} \right), \tag{5.16} \]

where the coefficients \(\nu_i^{(j)}[m]\), \(\mu_i^{(j)}[m,k]\) are computed from a system of quadratic algebraic equations [3, 7]. These quantities can also be computed approximately by the numerical solution of the Cauchy problem for a certain system of ordinary differential equations of Riccati type [7, 25].

Let us discuss the connection between the solutions (5.10) and (5.16) of problems 5.1 and \(5.1^0\) for large values of \(m\).

First of all, the following assertion is valid concerning the closeness of the motions \(x(t)\) and \(y^0(t)\) of systems (5.1) and (5.12) for \(u\equiv 0\). Let us compare the motion \(x(t)\) of system (5.1), generated by the initial state \(x(t_0+\vartheta)=z(\vartheta)\) \((-\tau\leq \vartheta\leq 0)\), with the motion \(\{y(t)\}_m\) of system (5.12) with the initial state

\[ y^0(t_0)=z^0=z(0),\qquad y^{(i)}(t_0)=z^{(i)}=m\int_{-\frac{i}{m}}^{-\frac{i-1}{m}} z(\vartheta)\,d\vartheta . \tag{5.17} \]

Under this comparison of \(x(t)\) and \(y^0(t)\), convergence of \(y^0(t)\) to \(x(t)\) as \(m\to\infty\) takes place, which is sufficiently uniform in \(z(\vartheta)\). Namely, let

\[ \|z[\vartheta]\| = \left[ \sum_{i=1}^{n} \left( z_i^2(0)+\int_{-\tau}^{0} z_i^2(\vartheta)\,d\vartheta \right) \right]^{1/2}. \tag{5.18} \]

Then for any numbers \(\varepsilon>0\) and \(T>0\) one can indicate a number \(N\) such that

\[ \sum_{i=1}^{n}[y_i^0(t)-x_i(t)]^2\leq \varepsilon^2 \quad \text{for } t_0\leq t\leq t_0+T, \tag{5.19} \]

provided only that \(\|z(\vartheta)\|\leq 1\), \(m\geq N\), and the initial conditions \(\{y(t_0)\}_m\) are chosen in accordance with equality (5.17). The number \(N\) can be effectively estimated from the numbers \(\varepsilon, T\) and the matrices \(A, A_t\).

Relying on the closeness of \(y^0(t)\) to \(x(t)\) for large \(m\), one can prove a certain convergence of the solutions (5.16) to (5.10). The following assertion is valid.

Theorem 5.1. Let the stabilizability criterion for system (5.1) be satisfied. Then for any number \(\varepsilon>0\) one can indicate a number \(N\) such that, for all values \(m\geq N\), system (5.12) is stabilizable and the inequalities

\[ |\xi^0[z(\vartheta)]-\eta^0[\{z\}_m]|\leq \varepsilon\|z(\vartheta)\|, \tag{5.20} \]

\[ |I[t_0,z(\vartheta),\xi^0]-I_m[t_0,\{z\}_m,\eta^0]|\leq \varepsilon\|z(\vartheta)\|^2, \tag{5.21} \]

hold, provided only that the initial conditions \(\{z\}_m\) and \(z(\vartheta)\) are connected by the equalities (5.17).

The relation between the quantities \(\nu_i^{(j)}[m]\), \(\nu_i^{(j)}\), \(\mu_i^{(j)}[m,k]\), and \(\mu_i^{(j)}(\vartheta)\) in (5.10) and (5.16) can be described and estimated in the following way. Consider the piecewise-constant functions

\[ \mu_i^{(j)}[m,\vartheta]=m\mu_i^{(j)}[m,k]\quad \text{for}\quad -\frac{k}{m}\leqslant \vartheta<-\frac{k-1}{m}. \tag{5.22} \]

Then, as \(m\to\infty\), the functions \(\mu_i^{(j)}[m,\vartheta]\) converge uniformly to the functions \(\mu_i^{(j)}(\vartheta)\), and the quantities \(\nu_i^{(j)}[m]\) converge to the quantities \(\nu_i^{(j)}\).

The results formulated above justify the approximation of problem 5.1 by problem \(5.1^\circ\). Moreover, it also follows that, for large values of \(m\), we obtain motions of system (5.1) close to optimal ones if, in the law of regulation of system (5.1), we introduce the control (5.11), where, however, instead of the exact solutions \(\nu_i\) and \(\mu_i(\vartheta)\) of problem 5.1, we use their approximate values \(\nu_i[m]\) and the functions \(\mu_i[m,\vartheta]\), obtained by rule (5.22) from the solutions \(\nu_i[m]\) and \(\mu_i[m,k]\) of the auxiliary problem \(5.1^\circ\).

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Ural State University
named after A. M. Gorky

1. Report delivered at the First Belorussian Mathematical Conference, January 25–28, 1964. ↩