MATHEMATICS

L. N. ESHUKOV

ON CONDITIONS FOR THE EXISTENCE OF A SOLUTION OF ONE PROBLEM OF OPTIMAL CONTROL

(Presented by Academician L. S. Pontryagin on 10 VI 1968)

Consider a system of linear differential equations with constant coefficients (in vector notation)

\[ \frac{dx}{dt}=Ax+\sum_{s=1}^{r}u_s v^s(t) \quad \left(x=x(t)\subset R^n;\quad u_s\subset R^n,\ s=1,2,\ldots,r\right). \tag{1} \]

Here \(v^s(t)\), \(s=1,2,\ldots,r\), are functions piecewise continuous on any finite interval in \(t\) and satisfying the following additional condition: for arbitrary \(t=t_1\), the point
\[ u(t_1)=\sum_{s=1}^{r}u_s v^s(t_1) \]
belongs to some previously chosen domain \(\overline U\) of the subspace \(R^r\subset R^n\), defined by linearly independent vectors \(u_s\), \(s=1,2,\ldots,r\). We assume the control domain \(\overline U\) to be closed and to contain any sphere of sufficiently small radius with center at the origin.

In the usual problem of optimal speed \((^{1,2})\), it is required, for a point \(x_0\in R^n\), to indicate an admissible control \(v(t)=\{v^s(t)\}\), \(s=1,2,\ldots,r\), which transfers it along a trajectory of system (1) to the origin in the minimum time.

Below we consider a condition somewhat modified in comparison with the usual one. We shall say that the condition of general position of the control vectors \(u_s\), \(s=1,2,\ldots,r\), is satisfied if one can indicate such a set \(g\) of \(n\) pairs of natural numbers
\[ g=\{(m_i,s_i)\},\quad i=1,2,\ldots,n, \]
that the determinant composed of column vectors of the form \(A^{m_i}u_{s_i}\), completely specified by the set \(g\), exists and is not equal to zero. It can be shown that this condition is invariant with respect to nonsingular linear transformations of the space \(R^n\) or of the subspace \(R^r\).

Let us recall the well-known definition of controllable systems of differential equations. System (1) is called controllable if there exists a controllability domain \(Q\subset R^n\), open and containing the origin, from each point of which one can reach the origin in finite time, moving along a trajectory of system (1) under some admissible control \(v(t)\) \((^2)\). It can be shown that controllability of system (1) is determined by the choice of the space \(R^r\) and depends neither on the choice of the domain \(\overline U\), nor on the choice of the basis vectors \(u_s\), \(s=1,2,\ldots,r\), in this space. The controllability domain, of course, depends on the choice of the domain \(\overline U\).

Theorem. In order that the system of differential equations (1) be controllable, it is necessary and sufficient that the condition of general position of the control vectors \(u_s\), \(s=1,2,\ldots,r\), be satisfied,

\[ \left|\,A^{m_i}u_{s_i}\,\right|_{(g)}\ne 0. \tag{2} \]

Let us explain the proof of the theorem.

Sufficiency of the condition. Let us write the solution of system (1) for some \(t=h\):

\[ x(h)=e^{Ah}x_0+e^{Ah}\int_0^h e^{-A\tau}u(\tau)\,d\tau . \]

Requiring that, for an arbitrary \(t=h\), the solution fall at the origin of the coordinates, we obtain the equation

\[ -\int_0^h e^{-A\tau}u(\tau)\,d\tau=x_0 . \tag{3} \]

Divide the interval \([0,h]\) into \(n\) parts by as yet undetermined points \(t_i\), \(i=1,2,\ldots,n\), and put \(u(\tau)=a^i u_s\) in each of the corresponding partial intervals. The numbers \(a^i\), \(i=1,2,\ldots,n\), will for the time being be left undetermined, while the numbers \(s_i\), \(i=1,2,\ldots,n\), will be taken from the set \(g\). Writing in (3) the matrix function \(e^{-A\tau}\) in the form of a power series and integrating on each of the partial intervals, we obtain a linear algebraic system for finding the numbers \(a^i\), \(i=1,2,\ldots,n\),

\[ \sum_{i=1}^{n}\varphi_i(A)a^i u_{s_i}=x_0 . \]

Let us write the determinant of this system:

\[ \det \|\varphi_i(A)u_{s_i}\|=\sum_{\{g\}}\psi_g(t_1,\ldots,t_n). \]

By condition (2), at least one of the determinants \(d_g\) occurring in the right-hand side is different from zero. The polynomials \(\psi_g\) cannot vanish identically and will not be linearly dependent for different \(g\). Therefore one can choose numbers \(t_i\), \(i=1,2,\ldots,n\), from the interval \([0,h]\) such that the determinant of the indicated system does not vanish. Then, for any point \(x_0\) of some arbitrary open domain \(Q\), \(Q\ni 0\), having sufficiently small diameter, numbers \(a^i\) will be found such that the vectors \(a^i u_{s_i}\), \(i=1,2,\ldots,n\), will belong to the control region \(\overline U\). System (1) is controllable.

Necessity of the condition. By a linear transformation of system (1), reduce the matrix \(A\) to (lower) Jordan form. Introduce the notation: \(\lambda_j\), \(j=1,2,\ldots,\sigma\), are the distinct eigenvalues of the matrix \(A\); \(\sigma\) is their number; \(q_j\) is the number of elementary blocks of the form

\[ B_{jp}=B_{jp}(\lambda_j)= \begin{pmatrix} \lambda_j & 0 & \ldots & 0 & 0\\ 1 & \lambda_j & \ldots & 0 & 0\\ 0 & 0 & \ldots & 1 & \lambda_j \end{pmatrix}, \]

containing the eigenvalue \(\lambda_j\); \(l_{jp}\) is the number of rows (columns) in these blocks, \(p=1,2,\ldots,q_j\), \(j=1,2,\ldots,\sigma\). Combine the blocks into arrays (of the first kind) \(M_j^*=(B_{jp})\), \(p=1,2,\ldots,q_j\), corresponding to the eigenvalue \(\lambda_j\), \(j=1,2,\ldots,\sigma\). Within each array arrange the blocks according to the relation \(l_{jp}\ge l_{j,p+1}\), \(p=1,2,\ldots,q_j-1\). Arrange the sequence of arrays \(M_j^*\), \(j=1,2,\ldots,\sigma\), by means of the numbers \(q_j; l_{j1},\ldots,l_{jq_j}\): let \(q_j\ge q_{j+1}\) and \(l_{j1}\ge l_{j+1,1}\), if \(q_j=q_{j+1}\), etc. This ordering determines the order and numbering of the components of the vectors \(x\) and \(u_s\), \(s=1,2,\ldots,r\), of system (1).

From the components of the vectors \(u_s\), \(s=1,2,\ldots,r\), form the matrices

\[ U_j=\|u_s^{m_{jp}+1}\|,\qquad p=1,2,\ldots,q_j;\quad s=1,2,\ldots,r, \]

for all \(j=1,2,\ldots,\sigma\).

Here, the letters \(m_{jp},\ p=1,2,\ldots,q_j;\ j=1,2,\ldots,\sigma,\) denote the numbers of the rows preceding the cell \(B_{jp}\) in the matrix \(\hat A\). Let us find the numbers

\[ r_j=\operatorname{rank} U_j,\qquad j=1,2,\ldots,\sigma . \tag{4} \]

In what follows we shall study the conditions

\[ r_j=q_j,\qquad j=1,2,\ldots,\sigma . \tag{5} \]

In addition to the arrays \(M_j^*,\ j=1,2,\ldots,\sigma,\) we shall need arrays (of the second kind) \(M_p^{**}=(B_{jp}),\ j=1,2,\ldots,\sigma_p,\) constructed for all \(p=1,2,\ldots,q_1\). Here the numbers \(\sigma_p\) are determined from the conditions \(q_{\sigma_p}\ge p,\ q_{\sigma_p+1}<p\). We shall denote by the letters \(h_p\) the numbers of rows (columns) in the arrays \(M_p^{**},\ p=1,2,\ldots,q_1\).

Let us prove that conditions (5) are necessary conditions for controllability of system (1). Suppose that for some one \(j=1,2,\ldots,\sigma\) we obtain the inequality \(r_j<q_j\). From system (1) choose all equations with numbers \(m_{jp}+1,\ p=1,2,\ldots,q_j\). They form a subsystem (in vector form)

\[ \frac{d\hat x}{dt}=\lambda_j\hat x+\sum_{s=1}^{r}\hat u_s v^s(t). \]

Construct a nonsingular transformation \(\hat x=P\hat y\) of the corresponding subspace \(R^{q_j}\subset R^n\). As the first \(r_j\) columns of the matrix \(P\) take those \(r_j\) linearly independent vectors which are found among the vectors \(\hat u_s,\ s=1,2,\ldots,r,\) by the definition (4) of the numbers \(r_j,\ j=1,2,\ldots,\sigma\). Then the indicated subsystem in the new coordinates will have unit vectors as control vectors. The number of these vectors is less than the dimension of the space \(R^{q_j}\), \(r_j<q_j\). Therefore the last \(q_j-r_j\) equations will not contain controls and will have the form

\[ dy^{r_j+i}/dt=\lambda_j y^{r_j+i},\qquad i=1,2,\ldots,q_j-r_j. \]

This system has the trivial solution \(u^{r_j+i}=0,\ i=1,2,\ldots,q_j-r_j\). In the space \(R^n\) it forms a linear manifold \(L\) containing the origin. The solution of system (1), determined by an initial point \(x_0\notin L\), does not enter \(L\) in finite time under any admissible control. The controllability domain cannot be constructed, and system (1) is uncontrollable.

We proceed to the proof of the assertion that fulfillment of condition (5) implies fulfillment of condition (2). If \(r_j=q_j,\ j=1,2,\ldots,\sigma,\) then the numbers \(s_{jp},\ p=1,2,\ldots,q_j,\) can be so chosen from the numbers \(s=1,2,\ldots,r\) that the conditions

\[ \tilde\Delta_{jp}=\left|u_{s_{jp}}^{m_{jk}+1}\right|_{p,k=1,2,\ldots,q}\ne 0 \tag{6} \]

will be satisfied for all \(q=1,2,\ldots,q_j\) and \(j=1,2,\ldots,\sigma\).

These sets of numbers \(s_{jp},\ p=1,2,\ldots,q_j,\) are different for different \(j=1,2,\ldots,\sigma\). We shall replace the vectors \(u_s,\ s=1,2,\ldots,r,\) by new vectors \(v_p,\ p=1,2,\ldots,r,\) taken in the form of linear combinations of the former vectors. We require that, for the components of the new vectors, the conditions

\[ \Delta_{jq}=\left|v_p^{m_{jk}+1}\right|_{p,k=1,2,\ldots,q}\ne 0 \tag{7} \]

be satisfied for all \(q=1,2,\ldots,q_j\) and \(j=1,2,\ldots,\sigma\) simultaneously. Put \(v_{1,p}=u_{s_{1p}},\ p=1,2,\ldots,q_1\). Conditions (7) are satisfied for \(j=1\) and \(q=1,2,\ldots,q_1\).

Suppose that the vectors \(v_{n,p},\ p=1,2,\ldots,q_1,\) have been constructed and that conditions (7) are satisfied for \(j=1,2,\ldots,h\) and \(q=1,2,\ldots,q_j\). Put \(v_{n+1,p}=v_{n,p}+\)

\(+ \alpha^{n+1}u_{s\,n+1,p},\ 1 \le p \le q_{n+1},\ v_{n+1,p}=v_{h,p},\ q_{n+1}<p \le q_1,\) where \(\alpha^{n+1}\) is an undetermined coefficient. Then, with the aid of conditions (6), one can choose the number \(a^{h+1}\) so that condition (7) will be satisfied for \(j=1,\ldots,h+1\) and \(q=1,\ldots,q_j\). For \(h=q_1\) put \(v_p=v_{q_1,p}\), \(p=1,2,\ldots,q_1\). For these vectors all conditions (7) are satisfied. It is not difficult to complete them to a basis in \(R^r\), if \(q_1<r\).

We shall first introduce the construction of the determinant \(d_q\) (2) by means of the arrays \(M_p^{**}\), \(p=1,2,\ldots,q_1\). To each such array \(M_p^{**}\) we assign a vector \(v_p\), \(p=1,2,\ldots,q_1\). Then to each column of this array we assign a vector of the form \(A^{k_p}v_p\) with increasing exponent \(k_p=0,1,\ldots,h_p-1\), where \(h_p\) is the number of columns in \(M_p^{**}\). Thus to each column of the matrix \(A\) there will correspond one vector of the form \(A^{k_p}v_p\). From these vectors we form a determinant, arranging them according to the ordering of the corresponding columns of the matrix \(A\). The order in which the vectors \(A^{k_p}v_p\) occur in the determinant obtained will correspond to the distribution of the columns of the matrix \(A\) among the arrays \(M_j^*\), and not among the arrays \(M_p^{**}\). We denote the determinant with this ordering of the columns by \(d\),

\[ d=\left|A^{k_p}v_p\right|. \tag{8} \]

Lemma. To each array \(M_j^*\), \(j=1,2,\ldots,\sigma\), assign the number

\[ d_j^*=\prod_{q=1}^{q_j}\Delta_{jq}^{\,l_{jq}-l_{j,q+1}} \]

(where \(l_{j,q_j+1}=0\) is put), and to each array \(M_p^{**}\), \(p=1,2,\ldots,q_1\), the number

\[ d_p^{**}= \prod_{j',\,j=1,\ j'>j}^{\sigma_p} (\lambda_{j'}-\lambda_j)^{-l_{j'p}\,l_{jp}} . \]

Then the determinant \(d\) (8) is computed by the formula

\[ d=\prod_{j=1}^{\sigma} d_j^* \prod_{p=1}^{q_1} d_p^{**}. \tag{9} \]

In proving this lemma, we replace columns of the form \(A^{k_p}v_p\) by columns of the form

\[ (A-\lambda_jE)^{s-1}\prod_{i=1}^{j-1}(A-\lambda_iE)^{l_{ip}}v_p, \qquad s=1,2,\ldots,l_{jp}, \]

for each column with number \(s\) from the cell \(B_{jp}\). This replacement does not change the value of the determinant \(d\). Then all elements standing to the right and above the arrays \(M_j^*\) are annihilated. After this we compute the determinants corresponding to the arrays \(M_j^*\).

Using the lemma, it is easy to show that the determinant \(d\) (8) is different from zero. The theorem is proved.

The main theorem can also be formulated in the following form:

If the matrix \(A\) is reduced to Jordan form, then a necessary and sufficient condition for the controllability of system (1) is the fulfillment of conditions (5).

Received
10 VI 1967

REFERENCES

1. L. S. Pontryagin, V. G. Boltyanskii et al., Mathematical Theory of Optimal Processes, Moscow, 1961.
2. V. G. Boltyanskii, Mathematical Methods of Optimal Control, “Nauka,” 1966.