UDC 517.934

ON CONTROL OVER THE MOTION OF NONLINEAR SYSTEMS

E. G. ALBRECHT

The problem is considered of constructing a control action that brings a certain nonlinear controlled system to a prescribed motion under small initial disturbances.

§ 1. STATEMENT OF THE PROBLEM

Consider an automatic-control system whose behavior is described by the differential equations of the disturbed motion

\[ \frac{dx}{dt}=\varphi(x,t)+\psi(x,t)u. \tag{1.1} \]

Here \(x=\{x_1,\ldots,x_n\}\) is the \(n\)-dimensional vector of phase coordinates of the system; \(\varphi(x,t)\), \(\psi(x,t)\) are \(n\)-dimensional vector functions; \(u\) is a scalar function describing the control action.

We shall assume that \(\varphi(x,t)\), \(\psi(x,t)\) are continuous functions of time \(t\) on the interval \([0,T]\) and analytic functions of the variables \(x_i\) in some neighborhood of the point \(x=0\), i.e., that they expand in convergent power series

\[ \varphi(x,t)=\sum_{k=1}^{\infty}\varphi^{(k)}(x,t), \qquad \psi(x,t)=\sum_{k=0}^{\infty}\psi^{(k)}(x,t), \tag{1.2} \]

where the symbol \((k)\) denotes the order of the form.

The following two problems are considered in the paper.

Problem 1.1. It is required to find a control action \(u^0(t)\) that brings system (1.1) to the undisturbed motion \(x=0\) in a prescribed finite time \(T\).

Problem 1.1 does not have a unique solution. We shall seek two such solutions—we shall seek controls close to the optimal control under the conditions:

\[ \max_{\tau}\bigl(|u^0(\tau)| \ \text{for } \ 0\leq t_0\leq \tau\leq T\bigr)=\min_u, \tag{1} \]

\[ \int_{t_0}^{T}[u^0(\tau)]^2\,d\tau=\min_u, \tag{2} \]

The aim of the present paper is to prove the convergence of the iterative method for constructing the control action \(u(t)\), proposed in [1], for sufficiently small initial disturbances \(x_i^0\).

§ 2. Preliminary Remarks

Consider the system of equations of the first approximation for (1.1)

\[ \frac{dx}{dt}=A(t)x+b(t)u, \tag{2.1} \]

where \(A(t)\) is an \(n\times n\) matrix;
\(A(t)x=\{\varphi_1^{(1)}(x,t),\ldots,\varphi_n^{(1)}(x,t)\}\);
\(b(t)=\{\psi_1^{(0)}(t),\ldots,\psi_n^{(0)}(t)\}\).

We shall assume that system (2.1) is completely controllable [2—5]. For this it is sufficient that the following condition be satisfied.

Condition 2.1. Let the matrix \(A(t)\) and the vector \(b(t)\) be continuously differentiable \(n-1\) times. The vectors \(L_1(t),\ldots,L_n(t)\) are linearly independent at least for one \(t^*\) from the interval \((t_0,T)\), \(t_0\geq 0\), i.e.

\[ \sum_{i=1}^{n}\lambda_i L_i(t^*)\ne 0 \quad \text{for} \quad \sum_{i=1}^{n}\lambda_i^2\ne 0, \]

where

\[ L_1(t)=b(t);\qquad L_{k+1}(t)=\frac{dL_k(t)}{dt}-A(t)L_k(t) \quad (k=1,\ldots,n-1). \]

We shall say that condition 2.1 is satisfied uniformly in \(t\) if the determinant \(\Delta(t)\) of the matrix \(\|L_1(t),\ldots,L_n(t)\|\) satisfies the inequality \(|\Delta(t^*)|\geq \varepsilon>0\) for each \(t^*\) from condition 2.1 for all \(t_0\geq 0\).

Let us now consider the following problem.

Problem 2.1. It is required to find a control \(u^0(t)\) which transfers the system of the first approximation (2.1) from the initial state \(x(t_0)=x^0\) to the terminal state \(x(T)=0\), under the condition that

\[ \max_{\tau}\bigl(|u^0(\tau)| \quad \text{for} \quad 0\leq t_0\leq \tau\leq T\bigr)=\min_u . \]

To solve Problem 2.1 it is necessary [2, 3] to find

\[ \beta=\min \int_{t_0}^{T}\left|\sum_{i=1}^{n} l_i h_i(\tau,t_0)\right|\,d\tau \tag{2.2} \]

under the condition

\[ \sum_{i=1}^{n} l_i c_i=1, \tag{2.3} \]

where \(h(\tau,t_0)=F(T,t_0)F^{-1}(\tau,t_0)b(\tau)\);
\(c=-F(T,t_0)x^0\); \(F(t,t_0)\) is the fundamental matrix of system (2.1) for \(u\equiv 0\), \(F(t_0,t_0)=E\).

The optimal control \(u^0(\tau)\) solving Problem 2.1 is determined by the relation

\[ u^0(\tau)=a^0(x^0,t_0)\operatorname{sign} \left(\sum_{i=1}^{n} l_i^0 h_i(\tau,t_0)\right). \tag{2.4} \]

Here \(a^0=\dfrac{1}{\beta^0}\), and \(\beta^0,l_i^0\) are solutions of the problem (2.2), (2.3) for the conditional extremum. This problem has a solution if system (2.1) is completely controllable [2—5], i.e. if condition 2.1 is satisfied.

Let the initial data \(x_i^0\) be such that \(\sum_{i=1}^n c_i^2 \ne 0\). Suppose, for definiteness, that \(c_n \ne 0\). Then from (2.3) it follows that

\[ l_n=\frac{1}{c_n}\left(1-\sum_{i=1}^{n-1} l_i c_i\right). \tag{2.5} \]

In this case the problem (2.2), (2.3) of a conditional extremum reduces to the problem of finding the minimum of the function

\[ \beta_1(l_1,\ldots,l_{n-1})= \int_{t_0}^{T} \left| \sum_{i=1}^{n-1} l_i h_i(\tau,t_0) + \frac{1}{c_n} \left( 1-\sum_{i=1}^{n-1} l_i c_i \right) h_n(\tau,t_0) \right|\,d\tau . \tag{2.6} \]

Therefore the numbers \(l_1^0,\ldots,l_{n-1}^0\) will satisfy the system of equations

\[ \frac{\partial \beta_1}{\partial l_i} = \int_{t_0}^{T} \left[ h_i(\tau,t_0)-\frac{c_i}{c_n}h_n(\tau,t_0) \right] \operatorname{sign} \left( \sum_{i=1}^{n-1} l_i h_i(\tau,t_0) + \right. \]

\[ \left. +\frac{1}{c_n} \left( 1-\sum_{i=1}^{n-1} l_i c_i \right) h_n(\tau,t_0) \right)\,d\tau =0 \quad (i=1,\ldots,n-1). \tag{2.7} \]

In what follows we shall assume that the initial data \(x_i^0\) are such that the following condition is satisfied.

Condition 2.2. The graph of the function

\[ h^0(\tau)=\sum_{i=1}^{n} l_i^0 h_i(\tau,t_0) \]

intersects the \(\tau\)-axis at angles different from zero. The Jacobian

\[ \partial\left( \frac{\partial \beta_1}{\partial l_1},\ldots, \frac{\partial \beta_1}{\partial l_{n-1}} \right) /\partial(l_1,\ldots,l_{n-1}) \]

for \(l_1=l_1^0,\ldots,l_{n-1}=l_{n-1}^0\) is different from zero, i.e.,

\[ \det\left\| \frac{\partial^2 \beta_1}{\partial l_i\,\partial l_k} \right\|_{1}^{\,n-1} \ne 0 \quad \text{for } \quad l_1=l_1^0,\ldots,l_{n-1}=l_{n-1}^0 . \]

Here

\[ \frac{\partial^2 \beta_1}{\partial l_i\,\partial l_k} = \sum_{j=1}^{s} \left[ h_i(\tau_j,t_0)-\frac{c_i}{c_n}h_n(\tau_j,t_0) \right] \frac{\partial \tau_j}{\partial l_k}. \]

The symbols \(\tau_j=\tau_j(l_1,\ldots,l_n)\) \((j=1,\ldots,s)\) denote the instants of time at which the function

\[ \sum_{i=1}^{n} l_i h_i(\tau,t_0) \]

vanishes. In this case, when Condition 2.2 is fulfilled, the derivatives

\[ \frac{\partial \tau_j}{\partial l_k} \quad \text{for } \quad l_1=l_1^0,\ldots,l_n=l_n^0 \]

exist

\[ (j=1,\ldots,s;\quad k=1,\ldots,n). \]

From condition 2.2 and from the theorem on implicit functions it follows that the numbers \(l_i^0\) will be continuously differentiable functions of the quantities \(c_1,\ldots,c_n\). Therefore, under small changes \(\Delta x_i^0\) of the quantities \(x_i^0\), or, what is the same, under small changes \(\Delta c_i\) of the quantities \(c_i\), there will occur small changes of the quantities \(l_i^0\), and the estimates

\[ |\Delta l_i^0|\leq N_1\|\Delta c\|,\qquad |\Delta a^0|\leq N_2\|\Delta c\|. \tag{2.8} \]

will hold.

Consequently, for all \(t\), with the exception of a set \(Q\) of values of \(t\), whose measure \(\mu(Q)\) satisfies the inequality

\[ \mu(Q)\leq N_3\|\Delta c\|, \tag{2.9} \]

the following inequality will hold:

\[ |\Delta u^0(t)|\leq N_4\|\Delta c\|, \tag{2.10} \]

where \(N_1,N_2,N_3,N_4\) are positive constants and \(\|\Delta c\|\) denotes the Euclidean norm of the vector \(\Delta c\). In general, everywhere in what follows, by the norm \(\|x\|\) of a vector \(x\) we shall mean the Euclidean norm, i.e.,

\[ \|x\|=\left(\sum_{i=1}^n x_i^2\right)^{1/2}. \]

Let us give an example for which the conditions 2.2 formulated above are fulfilled. Let

\[ \frac{dx}{dt}=y,\qquad \frac{dy}{dt}=u \tag{2.11} \]

and \(x(0)=1,\ y(0)=0\) be the initial state and \(x(1)=y(1)=0\) the terminal state of system (2.11). In this case we have

\[ h_1(\tau)=1-\tau,\qquad h_2(\tau)=1,\qquad c_1=1,\qquad c_2=0 \]

and

\[ \beta=\min \int_0^1 |l_1(1-\tau)+l_2|\,d\tau \quad \text{for } \quad l_1c_1+l_2c_2=l_1=1, \]

i.e.,

\[ \beta=\min_{l_2}\int_0^1 |1-\tau+l_2|\,d\tau, \]

then

\[ l_2^0=-\frac{1}{2}\quad \text{and}\quad h^0(\tau)=\frac{1}{2}-\tau. \]

The function \(h^0(\tau)\) vanishes at \(\tau=\tau_1=\frac{1}{2}\), and

\[ \frac{d^2\beta}{dl_2^2} = \left.\frac{d\tau_1}{dl_2}\right|_{l_2=l_2^0} =1\ne 0. \]

§ 3. METHOD OF SOLVING PROBLEM 1.1

Consider case 1.1(1). Let \(u_1(x^0,t_0,t)\) be the optimal control solving problem 2.1 for the first-approximation system,

and \(x^{(1)}(x^0,t_0,t)_{(2.1)}\) is the motion of system (2.1) for \(u=u_1(x^0,t_0,t)\). In this case the control \(u_1(x^0,t_0,t)\) satisfies the estimate

\[ |u_1(x^0,t_0,t)| \leq D_1\|x^0\|\qquad (D_1=\mathrm{const}). \tag{3.1} \]

In view of (3.1), the motion \(x^{(1)}(x^0,t_0,t)_{(2.1)}\) of the first-approximation system satisfies the inequality

\[ \|x^{(1)}(x^0,t_0,t)_{(2.1)}\| \leq N_5\|x^0\|\qquad (N_5=\mathrm{const}). \tag{3.2} \]

For \(u=u_1(x^0,t_0,t)\), the motion \(x^{(1)}(x^0,t_0,t)_{(1.1)}\) of the nonlinear system (1.1) is brought to the state [6] \(x^{(1)}(x^0,t_0,T)_{(1.1)}=y^{(2)}(T)\), and, with accuracy up to terms of second order of smallness in \(\|x^0\|\), we have

\[ y^{(2)}(t)=\int_{t_0}^{t} F(t,t_0)F^{-1}(\tau,t_0)r^{(2)}(\tau)\,d\tau, \tag{3.3} \]

where

\[ r^{(2)}(\tau)=\varphi^{(2)}\bigl(x^{(1)}(x^0,t_0,\tau)_{(2.1)},\tau\bigr) +\psi^{(1)}\bigl(x^{(1)}(x^0,t_0,\tau)_{(2.1)},\tau\bigr)u_1(x^0,t_0,\tau). \tag{3.4} \]

It follows from (3.1)—(3.4) that

\[ \|y^{(2)}(t)\|\leq M_2\|x^0\|^2\qquad (M_2=\mathrm{const}). \tag{3.5} \]

We now again consider problem 2.1 in the first approximation, concerning bringing system (2.1) not to the point \(x(T)=0\), but to the point \(x(T)=-y^{(2)}(T)\). Let \(u_2(x^0,t_0,t)\) be the solution of this problem. In this case \(c=-F(T,t_0)x^0-y^{(2)}(T)\), i.e. \(\Delta c^{(2)}=-y^{(2)}(T)\), and, in view of (2.8)—(2.10), (3.5), the inequality

\[ |\Delta u_2|=|u_2(x^0,t_0,t)-u_1(x^0,t_0,t)| \leq N_4\|y^{(2)}(T)\|\leq D_2\|x^0\|^2 \tag{3.6} \]

will hold for all \(t\), except for a set \(Q_2\) of values of \(t\), whose measure \(\mu(Q_2)\) satisfies the inequality

\[ \mu(Q_2)\leq N_3\|y^{(2)}(T)\|\leq \frac{N_3}{N_4}D_2\|x^0\|^2. \tag{3.7} \]

Consider the motion \(x^{(2)}(x^0,t_0,t)_{(1.1)}\) of system (1.1), corresponding to the control \(u=u_2(x^0,t_0,t)\). With accuracy up to terms of second order in \(\|x^0\|\), we obtain that \(x^{(2)}(x^0,t_0,T)_{(1.1)}=0\). If now terms of third order in \(\|x^0\|\) are taken into account, then \(x^{(2)}(x^0,t_0,T)_{(1.1)}=y^{(3)}(T)\), and

\[ y^{(3)}(t)=\int_{t_0}^{t} F(t,t_0)F^{-1}(\tau,t_0)r^{(3)}(\tau)\,d\tau, \tag{3.8} \]

where

\[ \begin{aligned} r^{(3)}(\tau)={}& \varphi^{(3)}\bigl(x^{(1)}(x^0,t_0,\tau)_{(2.1)},\tau\bigr) \\ &+\sum_{i=1}^{n} \frac{\partial \varphi^{(2)}\bigl(x^{(1)}(x^0,t_0,\tau)_{(2.1)},\tau\bigr)} {\partial x_i}\, \Delta x_i^{(2)}(x^0,t_0,\tau) \\ &+\psi^{(1)}\bigl(\Delta x^{(2)}(x^0,t_0,\tau),\tau\bigr)u_1(x^0,t_0,\tau) +\psi^{(1)}\bigl(x^{(1)}(x^0,t_0,\tau)_{(2.1)},\tau\bigr)\Delta u_2(\tau) \\ &+\psi^{(2)}\bigl(x^{(1)}(x^0,t_0,\tau)_{(2.1)},\tau\bigr)u_1(x^0,t_0,\tau). \end{aligned} \tag{3.9} \]

Here \(\Delta x^{(2)}(x^0,t_0,t)\) are the terms of second order in \(\|x^0\|\) in the solution of system (1.1). Moreover,

\[ \Delta x^{(2)}(x^0,t_0,t)=\int_{t_0}^{t} F(t,t_0)F^{-1}(\tau,t_0)\bigl(y^{(2)}(\tau)+b(\tau)\Delta u_2(\tau)\bigr)\,d\tau . \tag{3.10} \]

From (3.8)—(3.10) we have

\[ \|y^{(3)}(t)\|\leq M_3\|x^0\|^3 . \tag{3.11} \]

We again solve problem 2.1 in the first approximation, but now to the point \(x(T)=-y^{(2)}(T)-y^{(3)}(T)\). We denote the solution of this problem by \(u_3(x^0,t_0,t)\). Now \(\Delta c^{(3)}=-y^{(3)}(T)\), and by virtue of (2.8)—(2.10), (3.11) the estimates

\[ |\Delta u_3(t)|=|u_3(x^0,t_0,t)-u_2(x^0,t_0,t)|\leq N_4\|y^{(3)}(T)\|\leq D_3\|x^0\|^3, \tag{3.12} \]

\[ \mu(Q_3)\leq N_3\|y^{(3)}(T)\|\leq \frac{N_3}{N_4}D_3\|x^0\|^3 . \tag{3.13} \]

At the \(k\)-th step we obtain

\[ c=-F(T,t_0)x^0-\sum_{i=1}^{k}y^{(i)}(T),\qquad \Delta c^{(k)}=-y^{(k)}(T). \]

Consequently, the inequality

\[ |\Delta u_k(t)|=|u_k(x^0,t_0,t)-u_{k-1}(x^0,t_0,t)|\leq N_4\|y^{(k)}(T)\|\leq D_k\|x^0\|^k \tag{3.14} \]

will hold for all \(t\), except for a set \(Q_k\) of values of \(t\), whose measure \(\mu(Q_k)\) satisfies the inequality

\[ \mu(Q_k)\leq N_3\|y^{(k)}(T)\|\leq \frac{N_3}{N_4}D_k\|x^0\|^k . \tag{3.15} \]

Moreover,

\[ y^{(k)}(t)=\int_{t_0}^{t}F(t,t_0)F^{-1}(\tau,t_0)r^{(k)}(\tau)\,d\tau, \tag{3.16} \]

\[ \begin{aligned} r^{(k)}(\tau)=&\ \varphi^{(k)}\bigl(x^{(1)}(x^0,t_0,\tau)_{(2.1)},\tau\bigr)+ \\ &+\sum_{m=2}^{k-1}\sum_{l=1}^{m}\sum_{i_1,\ldots,i_l=1}^{n} \left( \frac{1}{l!}\, \frac{\partial^{(l)}\varphi^{(m)}\bigl(x^{(1)}(x^0,t_0,\tau)_{(2.1)},\tau\bigr)} {\partial x_{i_1}\ldots\partial x_{i_l}} \times \sum_{\substack{s_1,\ldots,s_l\geq 2\\ \sum s_i=k-m+l}} \Delta x_{i_1}^{(s_1)}\ldots \Delta x_{i_l}^{(s_l)} \right) \\ &+\sum_{s=1}^{k-1}\psi^{(s)}\bigl(x^{(1)}(x^0,t_0,\tau)_{(2.1)},\tau\bigr)\Delta u_{k-s}(\tau)+ \\ &+\sum_{s=1}^{k-1}\sum_{m=1}^{s-1}\sum_{l=1}^{m}\sum_{i_1,\ldots,i_l=1}^{n} \left( \frac{1}{l!}\, \frac{\partial^{(l)}\psi^{(m)}\bigl(x^{(1)}(x^0,t_0,\tau)_{(2.1)},\tau\bigr)} {\partial x_{i_1}\ldots\partial x_{i_l}} \times \sum_{\substack{s_1,\ldots,s_l\geq 2\\ \sum s_i=s-m+l}} \Delta x_{i_1}^{(s_1)}\ldots \Delta x_{i_l}^{(s_l)} \right)\Delta u_{k-s}(\tau). \end{aligned} \tag{3.17} \]

Here, for convenience of notation, the designation \(u_1(t)=\Delta u_1(t)\) has been adopted,

and \(\Delta x^{(s)}\) are the terms of order \(s\) in \(\|x^0\|\) in the solution of the nonlinear system (1.1), where

\[ \Delta x^{(s)}=\int_{t_0}^{t} F(t,t_0)F^{-1}(\tau,t_0)\bigl(y^{(s)}(\tau)+b(\tau)\Delta u_s(\tau)\bigr)\,d\tau . \tag{3.18} \]

Thus, the estimates (2.8)—(2.10), which follow from condition 2.2, ensure a reduction in the order of the vector \(y^{(k)}(T)\) with respect to \(\|x^0\|\), and consequently the control \(u(t)\) can be constructed so as to bring the system to the prescribed motion with any degree of accuracy with respect to \(\|x^0\|\).

§ 4. PROOF OF CONVERGENCE OF THE ITERATION PROCESS

Consider the series

\[ u(x^0,t_0,t)=u_1(x^0,t_0,t)+\sum_{k=2}^{\infty}\Delta u_k(x^0,t_0,t). \tag{4.1} \]

It follows from the arguments given in the preceding section that the control \(u(x^0,t_0,t)\), defined by relation (4.1), in the case of convergence of the series on the right-hand side of (4.1), carries the nonlinear system (1.1) from the initial state \(x(t_0)=x^0\) to the final state \(x(T)=0\). We shall show that the series (4.1) indeed converges in measure [7]. From the estimates (3.6), (3.7), (3.12), (3.13), (3.14), (3.15) it follows that, for this purpose, it is sufficient to show that the series [7, 8]

\[ v=\sum_{k=1}^{\infty} D_k\|x^0\|^k \tag{4.2} \]

converges.

For this we shall use the following estimates:

\[ \|\varphi(x,t)\|\leq \sum_{k=1}^{\infty} A_k\|x\|^k,\qquad \|\psi(x,t)\|\leq \sum_{k=0}^{\infty} B_k\|x\|^k, \tag{4.3} \]

\[ \left\|\frac{\partial \varphi^{(k)}}{\partial x_i}\right\|\leq kA_k\|x\|^{k-1},\qquad \left\|\frac{\partial \psi^{(k)}}{\partial x_i}\right\|\leq kB_k\|x\|^{k-1} \tag{4.4} \]

\[ (i=1,\ldots,n). \]

Here the series appearing on the right-hand side of the inequalities (4.3) converge in a sufficiently small neighborhood of the point \(x=0\).

Using the estimates (4.3), (4.4) and the relations (3.16), (3.17), we find a relation between the coefficients of the series (4.2)

\[ D_k=N^2\left( A_kN^k+ \sum_{m=2}^{k-1}\sum_{l=1}^{m} \sum_{\substack{\sum s_i=k-m+l\\ s_1,\ldots,s_l\geq 2}} C_m^l A_m D_{s_1}\cdots D_{s_l} +\right. \tag{4.5} \]

\[ \left. +\sum_{s=1}^{k-1} B_sD_{k-s}N^s +\sum_{s=1}^{k-1}\sum_{m=1}^{s-1}\sum_{l=1}^{m} \sum_{\substack{\sum s_i=s-m+l\\ s_1,\ldots,s_l\geq 2}} N^m C_m^l B_m D_{s_1}\cdots D_{s_l}\cdot D_{k-s} \right) \]

\[ (k=2,3,\ldots;\quad D_1=N), \]

where \(C_m^l\) is the number of combinations of \(m\) elements taken \(l\) at a time;

\[ N=1+\max\left\{N_1,\;N_2,\;N_3,\;N_4,\;N_5,\right. \]

\[ \left.\sup_{t,\tau}\left\|F(t,t_0)F^{-1}(\tau,t_0)\right\|T,\; \sup_{t,\tau}\left\|F(t,t_0)F^{-1}(\tau,t_0)b(\tau)\right\|T\right\}. \]

Consider the auxiliary equation

\[ z-N^2\left(\sum_{k=2}^{\infty} A_k N^k z^k + z\sum_{k=1}^{\infty} B_k N^k z^k\right)-N\vartheta=0. \tag{4.6} \]

It follows from the implicit function theorem that equation (4.6) has a solution \(z(\vartheta)\), analytic in \(\vartheta\) in a neighborhood of the point \(\vartheta=0\):

\[ z(\vartheta)=\sum_{k=1}^{\infty} E_k \vartheta^k. \tag{4.7} \]

Substituting (4.7) into (4.6) and equating the coefficients of like powers of \(\vartheta\), we obtain

\[ \begin{aligned} E_k=N^2\Bigg(&A_kN^{2k} +\sum_{m=2}^{k-1}\sum_{l=1}^{m} \sum_{\substack{s_1,\ldots,s_l\ge 2\\ \sum s_i=k-m+l}} C_m^l A_m N^{2m-l}E_{s_1}\cdots E_{s_l} \\ &+\sum_{s=1}^{k-1} B_sE_{k-s}N^{2s} +\sum_{s=1}^{k-1}\sum_{m=1}^{s-1}\sum_{l=1}^{m} \sum_{\substack{s_1,\ldots,s_l\ge 2\\ \sum s_i=s-m+l}} C_m^l B_m E_{k-s}N^{2m-l}E_{s_1}\cdots E_{s_l}\Bigg) \end{aligned} \tag{4.8} \]

\[ (k=2,3,\ldots;\; E_1=N). \]

Comparing (4.5) and (4.8), we see that the inequalities hold \((N>1)\)

\[ E_k>D_k\qquad (k=2,3,\ldots). \tag{4.9} \]

From inequalities (4.9), from the convergence of the series (4.7), and from the estimates (3.14), (3.15), it follows \([7,8]\) that the series (4.1) converges in measure.

Consequently, the functions \(u_k(x^0,t_0,t)\), which are partial sums of the series (4.1), converge in measure to the function \(u(x^0,t_0,t)\). Therefore the control \(u(x^0,t_0,t)\) is a relay sign-changing function of the form (2.4).

Let us now note that the control \(u(x^0,t_0,t)\) found in (4.1) is not an optimal control in the sense of Problem 1.1 (1). By construction, this control is optimal for the first-approximation system (2.1) when transferring it from the initial position to the terminal position

\[ x(T)=-\sum_{k=2}^{\infty} y^{(k)}(T). \tag{4.10} \]

From (3.14) and the convergence of the series (4.1) it follows that the series (4.10) converges and \(\|x(T)\|\) has second order of smallness with respect to \(\|x^0\|\).

Let us estimate the closeness of the control \(u(x^0,t_0,t)\) constructed by us to the optimal control \(u^0(x^0,t_0,t)\). Let

\[ p=\max_{\tau}\left(|u^0(x^0,t_0,\tau)|\right) \quad \text{for } \quad 0\leq t_0\leq \tau \leq T); \]

\[ q=\max_{\tau}\left(|u(x^0,t_0,\tau)|\right) \quad \text{for } \quad 0\leq t_0\leq \tau \leq T). \]

By virtue of the optimality of the control \(u^0(x^0,t_0,t)\), the inequality \(p\leq q\) is valid. For \(u=u^0(x^0,t_0,t)\), the linear system (2.1) will pass from the initial position \(x^0\) to the position \(\bar{x}(T)\), and the norm \(\|\bar{x}(T)\|\) will have second order of smallness with respect to \(\|x^0\|\). Let us now consider the optimal control \(u^*(x^0,t_0,t)\) for the first-approximation system (2.1), transferring it from the initial position \(x^0\) to the terminal position \(\bar{x}(T)\). Let

\[ p^*=\max_{\tau}\left(|u^*(x^0,t_0,\tau)|\right) \quad \text{for } \quad 0\leq t_0\leq \tau \leq T), \]

then \(p^*\leq p\). In this case \(\Delta c=\bar{x}(T)\), and therefore, by virtue of inequality (2.10), the quantity \(|p^*-p^{**}|\), where \(p^{**}=\max_{\tau}(|u_1(x^0,t_0,\tau)|\) for \(0\leq t_0\leq \tau \leq T)\), has second order of smallness with respect to \(\|x^0\|\). By construction, the quantity \(|q-p^{**}|\) also has second order of smallness with respect to \(\|x^0\|\). Then from the inequality \(p^*\leq p\leq q\) it follows that the quantity \(|q-p|\) also has second order of smallness with respect to \(\|x^0\|\). Consequently, the control \(u(x^0,t_0,t)\) (4.1) differs from the optimal control \(u^0(x^0,t_0,t)\) by a quantity of second order of smallness with respect to \(\|x^0\|\).

Thus, the following conclusion is valid:

Theorem 4.1. If the equations of the perturbed motion have the form (1.1) (where \(\varphi(x,t)\), \(\psi(x,t)\) are continuous functions of time \(t\) on the interval \([0,T]\) and analytic functions of \(x_i\) in some neighborhood of the coordinate origin \(x=0\)), and the first-approximation system (2.1) is completely controllable, then the nonlinear system (1.1) is also controllable for sufficiently small initial perturbations \(x_i^0\). If the initial data \(x_i^0\) are such that condition 2.2 is satisfied, then one can construct a control \(u(x^0,t_0,t)\) admissible in the sense of problem 1.1 (1), of the form (4.1)

\[ u(x^0,t_0,t)=\alpha(x^0,t_0)\operatorname{sign}\left(\sum_{i=1}^{n} l_i(x^0,t_0)h_i(t,t_0)\right). \]

Moreover, this control \(u(x^0,t_0,t)\) will differ from the optimal one by a quantity of second order of smallness with respect to \(\|x^0\|\).

§ 5. SOLUTION OF PROBLEM 1.1 (2)

Let us first consider the solution of problem 1.1 (2) in the first approximation.

Problem 5.1. It is required to find a control \(u^0(t)\) transferring system (2.1) from the initial state \(x_i(t_0)=x_i^0\) to the state \(x_i(T)=0\) \((i=1,\ldots,n)\), under the condition that

\[ \int_{t_0}^{T} [u^0(\tau)]^2\,d\tau=\min_{u^0}. \tag{5.1} \]

The solution of problem 5.1 has the form [9]

\[ u(x^0,t_0,t)=\lambda^2(x^0,t_0)\sum_{i=1}^{n} l_i^0 h_i(t,t_0), \tag{5.2} \]

where

\[ \lambda^{-2}(x^0,t_0)=\min \left\{\int_{t_0}^{T}\left(\sum_{i=1}^{n}l_i h_i(\tau,t_0)\right)^2 d\tau\right\} \tag{5.3} \]

under the condition

\[ \sum_{i=1}^{n} l_i c_i=1, \tag{5.4} \]

and \(l_i^0\) are the solutions of the problem (5.3), (5.4) for a conditional extremum. Carrying out the necessary computations, we obtain

\[ u(x^0,t_0,t)=\frac{1}{D(t_0)}\sum_{ij=1}^{n} h_j(t,t_0)D_{ij}(t_0)x_i^0, \]

where \(D(t_0)=\det \|a_{ij}\|_1^n \ne 0;\ D_{ij}\) are the cofactors in \(D\);

\[ a_{ij}=\int_{t_0}^{T} h_i(\tau,t_0)h_j(\tau,t_0)d\tau \quad (i,j=1,\ldots,n). \]

Moreover, if condition 2.1 is satisfied uniformly in \(t\), then it can be verified that \(|D(t)|>\delta>0\) for \(0\le t\le T-\varepsilon\) \((\varepsilon>0)\).

The control \(u^0(x^0,t_0,t)\), which solves problem 5.1, is a continuous function of the quantities \(x_i^0\), or, what is the same, of the quantities \(c_i\). Consequently, for small changes \(\Delta c_i\) in the quantities \(c_i\), estimate (2.8) will hold for all \(t\in[t_0,T]\). Therefore, in order to construct an admissible control \(u(t)\) in the sense of problem 1.1 (2), one may apply the iterative method described in the preceding paragraphs. Only now the series (4.1) will converge not with respect to the norm, but uniformly in \(t\). By arguments analogous to those given in §§ 3—4, we obtain the following conclusion.

Theorem 5.1. If the equations of the perturbed motion have the form (1.1) (where \(\varphi(x,t)\), \(\psi(x,t)\) are continuous functions of time on the interval \([0,T]\) and analytic functions of the phase coordinates \(x_i\) in some neighborhood of the point \(x=0\)) and the first-approximation system (2.1) is completely controllable, then the nonlinear system (1.1) is also controllable for sufficiently small initial perturbations. Moreover, an admissible control \(u(x^0,t_0,t)\) can be constructed, in the sense of problem 1.1 (2), in the form of the series (4.1). The control \(u(x^0,t_0,t)\) (4.1) will differ from the optimal control by a quantity of second order of smallness in \(\|x^0\|\) and will be an analytic function of the initial perturbations \(x_i^0\) and a continuous function of time on the interval \([0,T]\).

Remark 5.1. The results obtained in this paragraph make it possible to solve the synthesis problem [3—5], or the problem of the analytic construction of a regulator, i.e., to construct the control \(u[x,t]\) as a function of the phase coordinates of the system \(x_i\) and time \(t\). For this purpose, in (4.1) one must put \(t_0=t,\ x^0=x\). If condition 2.1 is satisfied uniformly in \(t\), then the series (4.1) will converge uniformly in \(t\), \(0\le t_0\le t\le T-\varepsilon\) \((\varepsilon>0)\), in a sufficiently small neighborhood of the origin of coordinates \(x=0\). In this case the control \(u[x,t]\) will be an analytic function of the coordinates \(x_i\) and a continuous function of time \(t\), \(0\le t_0\le t\le T-\varepsilon\) \((\varepsilon>0)\).

The author expresses deep gratitude to N. N. Krasovskii for the formulation of the problem and for his comments.

References

1. Krasovskii N. N., Shelementyev G. S. PMM, vol. XXIX, no. 3, 1965.

2. Akhiezer N., Krein M. On certain questions in the theory of moments. GONTI, 1938.

3. Krasovskii N. N. On the theory of optimal control. PMM, vol. XXIII, no. 4, 1959.

4. Pontryagin L. S., Boltyanskii V. G., Gamkrelidze R. V., Mishchenko E. F. The Mathematical Theory of Optimal Processes. Fizmatgiz, 1961.

5. Krasovskii N. N. PMM, vol. XXVII, no. 4, 1963.

6. Lefschetz S. Geometric Theory of Differential Equations. IL, 1961.

7. Lyusternik L. A., Sobolev V. I. Elements of Functional Analysis. Gostekhizdat, 1951.

8. Fichtenholz G. M. Course of Differential and Integral Calculus, vol. II. Fizmatgiz, 1962.

9. Krasovskii N. N. PMM, vol. XXVI, no. 2, 1962.

Received by the editors
November 1, 1965

Ural State University