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On a Certain Optimal Problem with a Single Switching
S. T. Zavalishchin
- Let a system of differential equations be given:
\[ \dot{x}=Ax-ku, \tag{1} \]
where \(A\) is a constant \(n\times n\) matrix; \(k\) is an \(n\)-dimensional vector; \(u\) is a scalar function; \(|u|\leq 1\).
Let the functional be given:
\[ J=\int_0^\infty f(x)\,dt, \tag{2} \]
where
\[ f(x)=\frac{1}{2}(x,Fx) \]
is a positive-definite quadratic form.
Suppose that \(\operatorname{Re}\lambda_i(A)<0\). Then there exists a symmetric matrix \(V\) such that
\[ VA+A'V=-F \tag{3} \]
and
\[ v(x)=\frac{1}{2}(x,Vx) \]
is a positive-definite quadratic form.
Introduce the function
\[ z(x)=(l,x), \tag{4} \]
where \(l=Vk\).
Compute the total derivative of the function \(v(x)\) along the system (1), taking (3), (4) into account:
\[ \dot{v}=-f-zu. \tag{5} \]
Integrating relation (5) over the interval from \(0\) to \(\infty\) and using (2), we obtain \(J=v(x_0)-P\), where
\[ P=\int_0^\infty zu\,dt. \tag{6} \]
Thus, the problem of finding the control \(u\) minimizing the functional (2) is equivalent to the problem of finding a control \(u\) that gives a maximum to the functional (6).
In the paper [1] the incorrect conclusion was drawn that the control
\[ u=\operatorname{sign} z \tag{7} \]
solves the stated optimal problem.* In fact, it is obvious that the choice of control must be made with account taken of the preceding state of the system and of the preceding values of \(u\). It is of interest, however, to indicate the conditions under which the solution (7) nevertheless remains correct. This interest is due to the consideration that in this case the optimal control depends only on the current state of the system and is simple to implement; moreover, difficulties of a computational character disappear.
The problem, in this form and with the indication of the error of Chang Jen-wei, was posed by E. A. Barbashin. In the present note necessary and sufficient conditions are given whose fulfillment ensures the validity of (7). It should be noted that the optimal control (7) belongs to the class of special ones. In the paper [3] some constructions of controls of this type are given.
- We shall prove a lemma.
Lemma. Let the equation
\[ \dot{y}=-\lambda y-\mu u, \tag{8} \]
\[ \text{* The system (19) of the paper [1] is inconsistent.} \]
where \(\lambda,\mu>0\), \(|u|\leqslant 1\). Let an integral criterion also be given,
\[ P=\int_0^\infty y u\,dt, \tag{9} \]
then the control
\[ u=\operatorname{sign} y \tag{10} \]
delivers the maximum of (9).
Proof. We form Bellman’s equation [2]
\[ \max_u \left| yu-\frac{d\Psi}{dy}(\lambda y+\mu u)\right|=0. \tag{11} \]
Putting
\[ u=\operatorname{sign}\left(y-\mu\frac{d\Psi}{dy}\right), \tag{12} \]
we obtain, for the determination of \(\Psi\), two equations:
\[ \frac{d\Psi}{dy}=\frac{y}{\mu+\lambda y}\quad \text{for } u>0, \tag{13} \]
\[ \frac{d\Psi}{dy}=\frac{y}{\mu-\lambda y}\quad \text{for } u<0. \tag{14} \]
Solving equations (13), (14), we obtain
\[ \Psi=\Psi^+ \quad \text{for } u>0, \tag{15} \]
\[ \Psi=\Psi^- \quad \text{for } u<0. \tag{16} \]
Set
\[ \Psi= \begin{cases} \Psi^+, & y>0,\\ \Psi^-, & y<0. \end{cases} \tag{17} \]
Let us note that it is not difficult to verify the conditions of continuous differentiability of the function (17) at \(y=0\). Taking (17) into account, we shall have
\[ y-\mu\frac{d\Psi}{dy}= \begin{cases} \dfrac{\lambda y^2}{\mu+\lambda y}, & y>0,\\[1.2ex] -\dfrac{\lambda y^2}{\mu-\lambda y}, & y<0. \end{cases} \]
Hence
\[ u=\operatorname{sign}\left(y-\mu\frac{d\Psi}{dy}\right)=\operatorname{sign} y, \]
which completes the proof.
Theorem. The control (7) of system (1) delivers the minimum to the criterion (2) if and only if the surface \(z(x)=0\) is a stable integral manifold for the uncontrolled system (1) (\(u=0\)).
Let us prove the sufficiency of the conditions. From the fact that \(z(x)=0\) satisfies the conditions of the theorem it follows that
\[ A'\Pi=-\lambda \Pi,\quad \lambda>0. \tag{18} \]
Taking (18) into account, we compute the total derivative of the function (4) along the system (1)
\[ \dot{\Pi}=-\lambda z-\mu u, \tag{19} \]
where \(\mu=(\Pi,k)=(k,Vk)>0\).
It was noted above that the minimization problem (2) is equivalent to the maximization problem (6). This, taking into account equation (19) and the lemma, proves the sufficiency of the conditions of the theorem.
Let us prove the necessity of the conditions. Suppose that the control (7) is optimal, but the conditions of the theorem are not satisfied. It can be shown that the transition process is completed in the sliding mode in the strip \(|(A'\Pi,x)|\leqslant (\Pi,b)\) from the instant \(T_1(x_0)\). Then the control having the structure
\[ u=\operatorname{sign} z \quad \text{for } t<T_1;\qquad u=0 \quad \text{for } T_1\leqslant t<T_2;\qquad u=\operatorname{sign} z \quad \text{for } T_2<t, \]
where \(T_2 - T_1\) is so small that the tendency to zero of the solutions of system (1) is preserved, gives the criterion (6) a larger value than the control (7). The contradiction obtained proves the necessity of the conditions of the theorem.
Remark. Let us note the validity of the theorem in the case when \(\lambda = 0\), and the criterion is taken in the form (6). This follows from a direct analysis of expression (9) for \(y\) obtained from (8). We have
\[ y = y_0 - \mu J(t), \qquad J(t) = \int_0^t u(\tau)\,d\tau, \qquad P = y_0 J(\infty) - \mu \int_0^\infty u(t)J(t)\,dt . \tag{20} \]
In expression (20) the integral is integrated by parts, after which we arrive at the relation
\[ P = y_0 J(\infty) - \frac{1}{2}\mu J^2(\infty). \tag{21} \]
Expression (21) has a maximum when
\[ J(\infty) = \frac{y_0}{\mu}. \tag{22} \]
This relation is satisfied by the control (10); in this case the phase point reaches the origin in the time
\[ T_0 = \frac{y_0}{\mu}. \]
- As an example of the application of the theorem, consider the problem of finding all vectors \(k\) and corresponding controls \(u\) that minimize the functional (2) in the case when \(f(x)=x^2\), \(A=\operatorname{diag}(-\lambda_1,\ldots,-\lambda_n)\). From relations (3), (4) we find
\[ V=\operatorname{diag}\left(\frac{1}{\lambda_1},\ldots,\frac{1}{\lambda_n}\right), \qquad \Pi=\left(\frac{k_1}{\lambda_1},\ldots,\frac{k_n}{\lambda_n}\right). \]
The desired vectors \(k\) satisfy condition (18), whence we obtain the relations for determining \(k\):
\[ k_i\frac{\lambda_i-\lambda}{\lambda_i}=0 \quad (i=1,\ldots,n). \tag{23} \]
In (23), \(\lambda=\lambda_j\) \((1\le j\le n)\), therefore the only solutions of (23) will be
\[ k^{(j)}=(\delta_{1j}k_1,\ldots,\delta_{nj}k_n)\quad (j=1,\ldots,n), \tag{24} \]
where \(\delta_{ij}\) is the Kronecker symbol. The controls corresponding to (24) and solving the minimization problem (2) are determined by the formulas
\[ u^{(j)}=\operatorname{sign} x_j \quad (j=1,\ldots,n). \]
References
- Chzhan Zhen-wei. Automation and Remote Control, 22, No. 12, 1601–1607, 1961.
- Bellman R. Dynamic Programming. IL, 1960.
- Uongkhem, Johnson. Optimal relay control with a quadratic performance index. Transactions of the American Society of Mechanical Engineers. Series D, No. 1. Theoretical Foundations of Engineering Calculations. 1964.
Received by the editors
May 18, 1965
Sverdlovsk Branch of the Mathematical Institute
named after V. A. Steklov