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UDC 517.917
ON THE CONTROL OF THE MOTION OF A QUASILINEAR SYSTEM
A. I. Subbotin
§ 1. FORMULATION OF THE PROBLEM
Let the behavior of the controlled system be described by the differential equations of the perturbed motion
\[ \frac{dx}{dt}=A(t)x+\mu f(x,t)+b(t)u, \tag{1.1} \]
where
\[ x=\{x_1,x_2,\ldots,x_n\}, \qquad f(x,t)=\{f_1(x,t),f_2(x,t),\ldots,f_n(x,t)\} \]
and \(b(t)=\{b_1(t),b_2(t),\ldots,b_n(t)\}\) are \(n\)-dimensional vectors; \(A(t)\) is an \(n\times n\)-matrix; \(u(t)\) is a scalar function; \(\mu\) is a small parameter.
We shall assume that the following conditions are satisfied:
I. \(f(x,t)\) satisfies, in some simply connected closed domain \(\Gamma\), the Cauchy–Lipschitz condition, i.e.
\[ \|f(x+\Delta x,t)-f(x,t)\|\leq P\|\Delta x\|,\quad (x,t)\in\Gamma,\ (x+\Delta x,t)\in\Gamma, \]
where \(P\) is a positive constant.
Here and everywhere below, by the norm \(\|x\|\) of a vector \(x\) we shall mean the Euclidean norm, i.e.
\[ \|x\|=\left[\sum_{i=1}^{n}x_i^2\right]^{1/2}. \]
II. The system (1.1) with \(\mu=0\) is controllable [1]. For this it is sufficient to assume [2] that the elements of the matrix \(A(t)\) and of the vector \(b(t)\) have derivatives up to order \(n-1\), inclusive, and that there exists at least one point \(t^*\in[t_0,T]\) such that the vectors \(L_1(t^*)\), \(L_2(t^*)\), \(\ldots\), \(L_n(t^*)\) are linearly independent, where
\[ L_1(t)=b(t),\qquad L_{k+1}(t)=\frac{dL_k(t)}{dt}-A(t)L_k(t),\quad k=1,2,\ldots,n-1. \]
We shall consider the following problem on optimal control [3].
Problem 1.1. It is required to find a control \(u^0(t)\) that transfers the system (1.1) from the initial state \(x(t_0)=X^0\), in a prescribed time \(T-t_0\), to the terminal state \(x(T)=0\), under the condition that
\[ \int_{t_0}^{T}[u^0(t)]^2\,dt=\min_u. \]
The aim of this paper is to describe a method of successive approximations for determining the control \(u(t)\) for sufficiently small values of the parameter \(\mu\). In accordance with the formulation of Problem 1.1, it should be assumed that
admissible controls \(u(t)\), which are elements of the space \(L_2[t_0,T]\), where the norm \(\|u(t)\|_{L_2}\) is defined by the equality
\[ \|u(t)\|_{L_2}=\left[\int_{t_0}^{T} u^2(t)\,dt\right]^{\frac12}. \]
We shall seek a control \(u(t)\) close to the optimal one in the following sense:
\[ \|u(t)-u^0(t)\|_{L_2}=O(\mu^{1/2}). \]
Here the symbol \(O(x)\) denotes a quantity of order \(Kx\). The control \(u(t)\) is constructed for initial disturbances \(x(t_0)=X^0\) for which the optimal motion of system (1.1) for \(\mu=0\) is entirely contained in the domain \(\Gamma\) together with some \(\varepsilon\)-neighborhood \((\varepsilon=\varepsilon(\mu))\).
§ 2. METHOD FOR SOLVING PROBLEM 1.1
Let us first consider problem 1.1 for \(\mu=0\). The solution of this problem exists if condition II is satisfied and has the form [1—3]:
\[ u_1^0(X^0,t_0,t)=\gamma^{-2}\sum_{i=1}^{n}\lambda_i^0 h_i(t^0,t), \tag{2.1} \]
where
\[ \gamma^2=\min \int_{t_0}^{T}\left(\sum_{i=1}^{n}\lambda_i h_i(t^0,t)\right)^2\,dt \]
under the condition
\[ \sum_{i=1}^{n}\lambda_i C_i=1, \tag{2.2} \]
and the numbers \(\lambda_i^0\) are the solution of problem (2.2) for a conditional extremum.
In (2.1) and (2.2), \(C=-F(t_0,T)X^0\); \(F(t_0,t)\) is the fundamental matrix of system (1.1) for \(\mu=0\), \(F(t_0,t_0)=E\); \(h(t_0,t)=F^{-1}(t_0,t)b(t)\). Carrying out the necessary computations, we obtain
\[ u_1^0(X^0,t^0,t)=\frac{-1}{D(t_0)}\sum_{i,k=1}^{n}D_{ik}(t_0)h_k(t_0,t)X_i^0, \tag{2.3} \]
where \(D(t_0)=\det\|\alpha_{ik}\|_1^n\), \(D_{ik}(t_0)\) are the cofactors in \(D(t_0)\);
\[ \alpha_{ik}=\int_{t_0}^{T} h_i(t_0,t)h_k(t_0,t)\,dt. \]
Consider an auxiliary problem.
Problem 2.1. For the system
\[ \frac{dy}{dt}=A(t)y+\mu g^{(i)}(t)+b(t)u_i \qquad (i=2,3,\ldots), \tag{2.4} \]
where \(g^{(i)}(t)\) is a known vector-function of time, it is required to find a control \(u_i^0(t)\) that transfers system (2.4) from the initial state \(y(t_0)=0\) to the terminal state \(y(T)=0\), under the condition that
ON THE CONTROL OF MOTION OF A QUASILINEAR SYSTEM
\[ \int_{t_0}^{T} [u_i^0(t)]^2\,dt=\min_{u_i}. \]
It follows from (2.3) that the solution of problem (2.1) has the form
\[ u_i^0(t_0,t)=\frac{1}{D(t_0)}\sum_{j,k=1}^{n}D_{jk}(t_0)h_k(t_0,t)\Delta C_j^{(i)}(T), \tag{2.5} \]
where
\[ \Delta C^{(i)}(t)=-\mu\int_{t_0}^{t}F^{-1}(t_0,\tau)g^{(i)}(\tau)\,d\tau . \tag{2.6} \]
Using (2.5) and (2.6), one can estimate \(u_i^0(t_0,t)\):
\[ \max_{t\in[t_0,T]}|u_i^0(t_0,t)|\leq \mu K_1 \max_{t\in[t_0,T]}\|g^{(i)}(t)\|. \tag{2.7} \]
For \(\mu\ne0\), we shall seek the solution of problem (1.1) in the form of the series
\[ u(X^0,t_0,t)=\sum_{i=1}^{\infty}u_i(t). \tag{2.8} \]
As \(u_1(t)\) we take the solution of problem 1.1 for \(\mu=0\), i.e., we set
\(u_1(t)=u_1^0(X^0,t_0,t)\) according to (2.3).
Let \(x^{(1)}(t)\) be the motion of system (1.1) for \(\mu=0\), \(u=u_1^0(X^0,t_0,t)\) and the initial condition \(x^{(1)}(t_0)=X^0\). We define the function \(u_2(t)\) as the solution of problem 2.1 for \(g^{(2)}(t)=f(x^{(1)}(t),t)\), i.e., we set \(u_2(t)=u_2^0(t_0,t)\).
Let \(x^{(2)}(t)\) be the solution of the system
\[ dx^{(2)}/dt=A(t)x^{(2)}+\mu g^{(2)}(t)+b(t)u_2^0(t_0,t) \]
with the initial condition \(x^{(2)}(t_0)=0\).
Suppose that \(u_{i-1}(t)\), \(x^{(i-1)}(t)\) are known. As \(u_i(t)\) we take the solution of problem 2.1 for
\[ g^{(i)}(t)=f\left(\sum_{j=1}^{i-1}x^{(j)}(t),t\right) -f\left(\sum_{j=1}^{i-2}x^{(j)}(t),t\right). \tag{2.9} \]
Let \(x^{(i)}(t)\) be the solution of the system
\[ \frac{dx^{(i)}}{dt}=A(t)x^{(i)}+\mu g^{(i)}(t)+b(t)u_i^0(t_0,t) \quad (i=2,3,\ldots) \tag{2.10} \]
with \(x^{(i)}(t_0)=0\).
Consider the series
\[ \bar{x}(t)=\sum_{i=1}^{\infty}x^{(i)}(t). \tag{2.11} \]
We shall show that the series (2.11), for sufficiently small values of the parameter \(\mu\), converges uniformly in \(t\) on \([t_0,T]\). Using (2.7), (2.9), and condition I, we have
\[ \max_{t\in[t_0,T]}|u_i^0(t_0,t)|\leq \mu K_2 \max_{t\in[t_0,T]}\|x^{(i-1)}(t)\|. \tag{2.12} \]
To determine sufficient conditions for convergence of the series (2.11), let us estimate
\(\max_{t\in[t_0,T]}\|x^{(i)}(t)\|\) in terms of \(\max_{t\in[t_0,T]}\|x^{(i-1)}(t)\|\). To this end, by Cauchy’s formula ([5], p. 135) we find the solution of system (2.10), for \(i \geqslant 2\), under the initial condition \(x^{(i)}(t_0)=0\):
\[ x^{(i)}(t)=-F(t_0,t)\Delta C^{(i)}(t)+F(t_0,t)\int_{t_0}^{t}h(t_0,\tau)u_i^0(t_0,\tau)\,d\tau . \tag{2.13} \]
From (2.13), (2.6), (2.9), (2.12) we obtain
\[ \max_{t\in[t_0,T]}\|x^{(i)}(t)\|\leqslant \mu K_3 \max_{t\in[t_0,T]}\|x^{(i-1)}(t)\|. \tag{2.14} \]
Thus, for \(\mu<\dfrac{1}{K_3}\), the series (2.11) converges uniformly in \(t\) on \([t_0,T]\); by (2.12), the series (2.8) will also converge uniformly on \([t_0,T]\).
We now show that the control (2.8) constructed above transfers system (1.1), along the trajectory (2.11), from the initial state \(\bar{x}(t_0)=X^0\) in the prescribed time \(T-t_0\) to the final state \(\bar{x}(T)=0\). To do this, we sum systems (2.10) over \(i\) from 2 to \(\infty\), and add to this sum system (1.1) for \(\mu=0\) and \(u=u_1^0(X^0,t_0,t)\). We obtain
\[ \sum_{i=1}^{\infty}\frac{dx^{(i)}}{dt} = A(t)\sum_{i=1}^{\infty}x^{(i)} +\mu\sum_{i=2}^{\infty}g^{(i)}(t) +b(t)\sum_{i=1}^{n}u_i(t). \tag{2.15} \]
By construction, we have
\[ \sum_{i=2}^{\infty}g^{(i)}=f(\bar{x}(t),t) \]
and, consequently,
\[ \frac{d\bar{x}(t)}{dt} = A(t)\bar{x}(t)+\mu f(\bar{x}(t),t)+b(t)u(X^0,t_0,t). \]
Therefore \(\bar{x}(t)\) is a solution of system (1.1), and, by Problems 1.1 and 2.1, we have
\(\bar{x}(t_0)=X^0,\ \bar{x}(T)=0\). Thus, the control \(u(X^0,t_0,t)\) (2.8) is an admissible control of system (1.1) in the sense of Problem 1.1.
§ 3. ESTIMATE OF THE CONTROL \(u(X^0,t_0,t)\)
We shall show that \(\|u(X^0,t_0,t)-u^0(t)\|_{L_2}=O(\mu^{1/2})\). For this purpose, consider the systems
\[ \frac{dx}{dt}=A(t)x+\mu f(x,t)+b(t)u, \tag{3.1} \]
\[ \frac{dy}{dt}=A(t)y+b(t)v, \tag{3.2} \]
where \(u(t)\) and \(v(t)\) are certain controls admissible in the sense of Problem 1.1, i.e., controls from \(L_2[t_0,T]\), which transfer systems (3.1), (3.2) from the initial state \(x(t_0)=X^0,\ y(t_0)=X^0\) in the prescribed time \(T-t_0\) to the final state \(x(T)=0,\ y(T)=0\). As before, \(u^0(t)\), \(v^0(t)\) will denote the optimal controls of systems (3.1), (3.2).
We shall prove the following lemmas:
Lemma 3.1. For any control \(v(t)\) there exists a control \(u^*(t)\) such that \(\|u^* - v\|_{L_2}=O(\mu)\); conversely, for any control \(u(t)\) there exists \(v^*(t)\) such that \(\|v^*-u\|_{L_2}=O(\mu)\).
Lemma 3.2. \(\|u^0\|_{L_2}-\|v^0\|_{L_2}=O(\mu)\).
Lemma 3.3. If \(\|v^0\|_{L_2}-\|v^*\|_{L_2}=O(\mu)\), then \(\|v^0-v^*\|_{L_2}=O(\mu^{1/2})\).
Proof of Lemma 3.1. Let \(v(t)\) be an arbitrary admissible control of system (3.2) in the sense of Problem 1.1. Substituting \(v(t)\) into (3.1) instead of \(u(t)\), by virtue of the continuous dependence of the solution of a system of differential equations on a parameter ([5], p. 178), we have: \(\|x(v,T)\|=O(\mu)\), where \(x(v,t)\) is the solution of the system
\[ \frac{dx}{dt}=A(t)x+\mu f(x,t)+b(t)v \tag{3.3} \]
with \(x(v,t^0)=X^0\).
Consider the system
\[ \frac{dz}{dt}=A(t)z+\mu\,[\,f(x(v,t)+z,t)-f(x(v,t),t)\,]+b(t)w. \tag{3.4} \]
Define \(w(t)\) as some control of system (3.4) that transfers this system from \(z(t_0)=0\) to \(z(T)=-x(v,T)=O(\mu)\). By the method described in § 2, one can construct \(w(t)\) so that
\[ \|w(t)\|_{L_2}=O(\mu). \tag{3.5} \]
Adding (3.4) and (3.3), we obtain that \(v+w=u^*\) carries out the motion of the system
\[ \frac{d(x+z)}{dt}=A(t)(x+z)+\mu f(x+z,t)+b(t)(v+w) \]
from \(x(t_0)+z(t_0)=X^0\) to \(x(T)+z(T)=0\). Consequently, \(u^*\) is an admissible control of system (3.1) in the sense of Problem 1.1. From (3.5) it follows that \(\|u^*-v\|_{L_2}=O(\mu)\). Thus, the first assertion of Lemma 3.1 is proved. The second assertion of this lemma is proved along the same lines as the first.
Proof of Lemma 3.2. Let \(\|u^0\|_{L_2}\geq \|v^0\|_{L_2}\). From Lemma 3.1 it follows that there exists \(u^*\) such that
\[ \|u^*\|_{L_2}-\|v^0\|_{L_2}=O(\mu), \qquad \text{but} \qquad \|u^*\|_{L_2}\geq \|u^0\|_{L_2}, \]
therefore
\[ 0\leq \|u^0\|_{L_2}-\|v^0\|_{L_2}\leq \|u^*\|_{L_2}-\|v^0\|_{L_2}=O(\mu). \]
Thus, \(\|u^0\|_{L_2}-\|v^0\|_{L_2}=O(\mu)\). If \(\|v^0\|_{L_2}\geq \|u^0\|_{L_2}\), then, using \(\|v^*\|_{L_2}-\|u^0\|_{L_2}=O(\mu)\), we again obtain that \(\|u^0\|_{L_2}-\|v^0\|_{L_2}=O(\mu)\).
We prove Lemma 3.3. Let \(\Delta v=v^*-v^0\). Thus,
\[ F(t_0,T)\left[X^0+\int_{t_0}^{T} h(t_0,t)v^0\,dt\right]=y(T)=0 \]
and
\[ F(t_0,T)\left[X^0+\int_{t_0}^{T} h(t_0,t)v^*\,dt\right]=y(T)=0, \]
hence
\[ \int_{t_0}^{T} h(t_0,t)\Delta v\,dt=0. \tag{3.6} \]
It follows from (2.1) that \(v^0=\sum_{i=1}^n a_i h_i(t_0,t)\), \(a_i=\mathrm{const}\), and therefore, by virtue of (3.6), we have
\[ \int_{t_0}^{T} v^0 \Delta v\,dt = \sum_{i=1}^{n} \left[ a_i \int_{t_0}^{T} h_i(t_0,t)\Delta v\,dt \right] =0. \tag{3.7} \]
From the fact that \(\|v^0\|_{L_2}-\|v^0+\Delta v\|_{L_2}=O(\mu)\), it follows that
\[ 2\int_{t_0}^{T} v^0 \Delta v\,dt + \int_{t_0}^{T} |\Delta v|^2\,dt = O(\mu). \tag{3.8} \]
From (3.8) and (3.7) we obtain
\[ \|\Delta v\|_{L_2}=O(\mu^{1/2}), \]
which was what had to be proved.
Let us now show that \(\|v^0-u^0\|_{L_2}=O(\mu^{1/2})\). From Lemma 3.1 we obtain
\[ \|u^0\|_{L_2}-\|v^*\|_{L_2}=O(\mu), \tag{3.9} \]
\[ \|v^0\|_{L_2}-\|u^*\|_{L_2}=O(\mu). \tag{3.10} \]
Subtracting (3.9) from (3.10) and using Lemma 3.2, we obtain
\[ \|v^*\|_{L_2}-\|u^*\|_{L_2}=O(\mu). \tag{3.11} \]
From (3.10) and (3.11) it follows that
\[ \|v^0\|_{L_2}-\|v^*\|_{L_2}=O(\mu). \tag{3.12} \]
According to Lemma 3.3 and (3.12),
\[ \|v^0-v^*\|_{L_2}=O(\mu^{1/2}). \tag{3.13} \]
Then
\[ \|v^0-v^*\|_{L_2}+\|v^*-u^0\|_{L_2}=O(\mu^{1/2}), \]
whence
\[ \|v^0-u^0\|_{L_2}=O(\mu^{1/2}). \]
By the construction of the control \(u(X^0,t_0,t)\), we have
\(\|u(X^0,t_0,t)-v^0\|_{L_2}=O(\mu)\); therefore, finally, we obtain
\(\|u(X^0,t_0,t)-u^0\|_{L_2}=O(\mu^{1/2})\).
The author thanks E. G. Albrecht for posing the problem and for valuable advice.
References
- Kalman R. E. Proceedings of the First IFAC Congress, 1. Publishing House of the Academy of Sciences of the USSR, 1961, pp. 521–530.
- Krasovskii N. N. PMM, vol. XXVI, no. 2, 218–232, 1962.
- Krasovskii N. N. PMM, vol. XXVII, no. 4, 641–663, 1963.
- Pontryagin L. S., Boltyanskii V. G., Gamkrelidze R. V., Mishchenko E. F. The Mathematical Theory of Optimal Processes. Fizmatgiz, 1961.
- Pontryagin L. S. Ordinary Differential Equations. “Nauka,” 1965.
- Fichtenholz G. M. Foundations of Mathematical Analysis, vol. II. Fizmatgiz, 1960.
Received by the editors
March 9, 1966
Ural State University
named after A. M. Gorky