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7URAL MATHEMATICAL JOURNAL, Vol. 3, No. 1, 2017
ASYMPTOTIC EXPANSION OF A SOLUTION FOR ONE SINGULARLY PERTURBED OPTIMAL
CONTROL PROBLEM IN Rn WITH A CONVEX INTEGRAL QUALITY INDEX
Alexander A. Shaburov
Institute of Natural Sciences and Mathematics, Ural Federal University,
Ekaterinburg, Russia, alexandershaburov@mail.ru
Abstract: The paper deals with the problem of optimal control with a convex integral quality index for a linear steady-state control system in the class of piecewise continuous controls with a smooth control constraints. In a general case, for solving such a problem, the Pontryagin maximum principle is applied as the necessary and sufficient optimum condition. In this work, we deduce an equation to which an initial vector of the conjugate system satisfies. Then, this equation is extended to the optimal control problem with the convex integral quality index for a linear system with a fast and slow variables. It is shown that the solution of the corresponding equation as e ^ 0 tends to the solution of an equation corresponding to the limit problem. The results received are applied to study of the problem which describes the motion of a material point in Rn for a fixed period of time. The asymptotics of the initial vector of the conjugate system that defines the type of optimal control is built. It is shown that the asymptotics is a power series of expansion.
Keywords: Optimal control, Singularly perturbed problems, Asymptotic expansion, Small parameter.
Introduction
The paper is devoted to studying the asymptotics of the initial vector of a conjugated state and an optimal value of the quality index in the optimal control problem [1]—[3] for a linear system with a fast and slow variables (see review [4]), convex integral quality index [3, Chapter 3], and smooth geometrical constraints for control.
Singularly perturbed problems of optimal control have been considered in different settings in [5]-[7].
The method of boundary function that was developed in [4, 10] allows effectively constructing an asymptotics of solutions for problems with an open control area and smooth controlling actions.
The solving of problems with a closed and bounded control area meets certain difficulties. That is why the problems with fast and slow variables and closed constraints for control have been studied to a less extent. A significant contribution to solving these problems was made by Dontchev and Kokotovic.
Problems of fast operation and terminal control with constraints for control in the form of a polygon are dealt with in [5, 7]. The structure of such optimal control is a relay function with values in the apexes of the polygon. No optimal control with constraints in the form of a sphere, which is a continuous function with a finite and countable number of discontinuity points, has been considered so far.
The asymptotics of solutions of the perturbed control problem was formulated differently in papers [7, 9].
In the present work, the basic equation for searching for the asymptotics of the initial vector of the conjugated state of the problem under consideration and optimal control is obtained. General relationships are applied to the case of the optimal control with a point of a small mass in an n-dimensional space under the action of a bounded force.
1. General statement of problem and condition for optimality
Let us consider a problem that belongs to the class of piecewise continuous controls - optimal control problem for a linear stationary system with a convex integral quality index:
z = Az + Bu, z(0) = z0, ||u(t)|| < 1, t e [0;T],
t (1.1)
J(u) = p(z(T)) + / ||u(t)||2 dt — min, 0
where z e R™, u e Rr, || • || is the Euclidean norm in Rr, A, B are constant matrices of the corresponding dimensional, and <^(-) is the convex function that is continuously differentiable in R™.
Note that in the considered convex integral quality index J, where the first term can be interpreted as a fine for the control error at a finite time instant T, whereas the second, as an account of an energy spent for the realization of control.
Condition 1. Let us assume that a pair (A, B) is quite controllable,
rank(B, AB,..., A™-1B) = n.
Under the conditions stated, the Pontryagin maximum principle in the problem (1.1) is the necessary and sufficient criterion of optimality. In this case, the problem has the unique solution [3, p. 3.5, Theorem 14]: if z, n is the unique solution to (1.1) and
n = -A*n, n(T) = -VV(z(T)), (1.2)
then the optimal control uo is determined from the maximum principle
-||u°(t)||2 + (B*n(t),uo(t)) = max ( - ||u||2 + (B*n(t),u)). (1.3)
IMI
Here (•, •) is the scalar product in Rr.
Calculating maximum in (1.3), we find
B*n(t) f 2, 0 < £ < 2,
u°(i> = • where SK):={ £> 2.
Note that the determination of function S(•) leads to the validity of inequality
w\ w2
Vwi,w2 e Rr Let A := n(T). Then
S(||wi |) S(|W2|)
^ ||wi - w2 |. (1.5)
t
n(t) = e-At(t-T)A, z(t) = eAtz0 + j eA(t-s)Bu°(s)ds
At a finite time instant t = T we have
T
AT 0 , eA(T-s)BB*eAt(T-s)A , AT "0 1 ' -ds.
z(T) = eAT z0 + J
S (HB*eA*(T-s)A|) 0
Replacing the variable t := T — s, we obtain
z(T) = eATzo + T ejATBB*eA*TA dT
z(T ) = e z + y s(\\fi*eA*tAy)
0
Thus, the following is valid:
Statement 1. Let condition 1 be valid, z(t), u(t) be a solution of the system from Problem (1.1), and n(t) be a solution of the system (1.2). Then z(t), n(t), u(t) is the solution of the maximum principle problem (1.1), (1.2), (1.3) if and only if when n(T) = A, u(t) is determined by the formula (1.4), and a vector A is the unique solution of equation
T
—A = MeT z° + / eAT B SlP^ 4 ^
0
Besides u(t) is the unique optimal control in the problem (1.1).
The vector A that satisfies the equation (1.6) will be called as a vector determining the optimal control in the problem (1.1).
Statement 2. Let uo(t) be the optimal control in (1.1). Then uo(t) is continuous on [0; T] and infinitely differentiable at points t such that ||B*eA (T-t ^A|| = 2. Here A is a vector determining the optimal control in problem (1.1).
Proof. The validity of statement follows from (1.4) and analytical form of the matrix exponent eAtt. □
2. Optimal control problem with fast and slow variables
Consider a particular case of problem (1.1), when the system under control contains fast and slow variables and the terminal part of the quality index depends only on slow variables:
' Xs = AUXe + AuVe + BlU, t £ [0,T], \\u\\ < 1,
£Ve = A2lX£ + A22Ve + B2U, X£ (0) = X0, y£ (0) = y0,
T (2.1) uitlir dt —>■ min,
J(u) := a(Xe(T))^y \\u(t)\\2 dt
0
where x £ Mra, y £ Mm, u £ Rr; Aj, Bi (i,j = 1,2) are the constant matrices of the corresponding dimensions, and a(-) is the convex function that is continuously differentiable in Rra.
Condition 2. All eigenvalues of matrix A22 have negative real parts.
For each fixed e > 0 the problem (2.1) coincides with the problem (1.1):
*t>- ( lit, )■ z0 = ( X0 ). * = ( e^21 & ). = ( e_BB2
n = n + m, p(z£) = a(xs).
As a limit problem for (2.1), the following problem is introduced
xo = A0x0 + B0u, t £ [0,T], ||u|| ^ 1,
Ao := An - A12A-21A2i, Bo := Bi - Ai2A-21B2, xo(0) = x0,
T
J(u) := a(xo(T)) + J ||u(t) ||2 dt ^ min 0
(2.2)
Condition 3. Pairs (A0,B0) and (A22,B2) are quite controllable.
If the Conditions 2-3 are satisfied, then there exists e0 > 0 such that the pair (Ae, B£) is quite controllable at any e : 0 < e ^ e0 [5, Theorem 1].
Note that since Vp(z£) = ^ Va(x£)^, then the vector Xs, which determines the optimal
control in the problem (2.1), has the form Ae = ^ ^ ^, le £ R™.
The vector le also will be called as determining the optimal control in problem (2.1). Let
.= ( We11(t) We12(t)
e £ := ( W21(t) W22(t) 1 ' (2.3)
then, by virtue of (2.3) the equation (1.6) transforms into
-le = W^ We11(T )x0 + W12(T )y0+
T- ,11(t)B. , (BKWHt))* +f-1B|(We12(t))*)le X (2'4)
+
I -1 \ (B;(Wen(t))* + e~1B* (We12(t)); )le
(We (t)B1 + e We (№j 5 (|(B ; (We11(t))*+ 6-1BI (W12(t))* )le
dt
Note that the optimal control u°(t) in the problem (2.1) is expressed through the vector le as follows:
o(T_t)= B (We11(t))*+ e-1Bl (We12(t))*)le (25)
Ue ( ) S( || (B\(We11(t))* + e~1B* (W£12(t))* )le I)' (.)
Theorem 1. Let the Conditions 2 and 3 be valid. Then le — l0 as e — +0, where le is the unique solution of the equation (2'4), and l0 is the unique solution of the equation
T
-l0 = ^(eA0Tx° + } eAo'BsBemiidt} (2.6)
Proof. It is known that the attainability set for the controllable system under control from (2.1) is uniformly bounded by the time instant T at e £ (0; e0] (see, for example, [6, theorem 3.1]). Hence, by virtue of (2.4) vectors {le} are also bounded at e £ (0;e0]. Therefore, to prove the theorem, it is sufficient to show that all partial limits {le} as e — +0 are equal to l0.
As follows from the A.B. Vasil'eva's results (see, for example [10, Chapter 3]) there is y > 0 such that
We11(t) = eAot + 0(e), W£12(t) = -eeAoiA12A-21 + 0(ee-Yt/e) + O(e2),
We21(t) = -A-21A21eAot + 0(e-Yt/e)+ O(e), We22(t) = 0(e-Yt/e). ^
Moreover, asymptotic estimates are uniform in t £ [0; T].
Hence, by virtue of (2.2) which determines the matrices A0 and B0 and by formulas (2.7) the expression standing Va for the formula (2.4) has the form
T
eAoTx° + O(e) + f (eAotBo + Ofe-^*) + O(e))_^B°eA°* + O(e-Y £) + O(e))l£_dt (2 8)
6 X + O(e) + J [e B0 + O(e )+ O(e)) 5 (\\(B*eAot + O(e-Y/) + O(e))l£\\) ^ (2.8)
Let us divide the integral from (2.8) into two terms f^ = fJ^ + . Then, taking into account
that the expression under integral is uniformly constrained and that O(e-l/^) = O(ea) as e ^ 0 for any a > 0, we obtain from (2.4) and (2.8)
T
-l• = + «M + + /eAB0sdt) ■
Let l be a partial limit of the vectors {l£} as e ^ +0, i.e. l£k ^ l for a certain {ek} so that ek ^ +0. Going to the limit as k ^ œ in (2.9) we obtain that l is the solution of (2.6). Because of the uniqueness of such a solution we have I = l0. □
The main problem for (2.1) is to find the complete asymptotic expansion in powers of small parameter e of the optimal control, optimal values of the quality index, and the optimal process. Formulas (2.5) and (1.5) show that if one manages to gain the complete asymptotic expansion of vector l£, which determines the optimal control in problem (2.1), this vector can be used for the asymptotic expansions of the above values as well.
3. Construction of complete asymptotic expansion of vector l£ for an optimal control problem with fast and slow variables
Consider a partial case of problem (2.1):
X£ = y£, t e [0,T], \\u\\ < 1,
eij£ = y £ + u, X£(0) = x0, y£(0) = y0,
t (3.1)
J(u) := 2\\X£(T)\\2 + / \\u(t)\\2 dt ^ min, 0
where x£, y£, u e Rra.
Problem (3.1) simulates a motion of a material point of small mass e > 0 with the coefficient of the medium resistance equals to 1 in the space Rra under action of the constrained control force u(t).
Here An = 0, Ai2 = I, A2i = 0, A22 = —I, B\ = 0, B2 = I, and 0 and I are the zero and the identity matrices of dimensional n x n, respectively. For the limit problem we have A0 = 0, B0 = I and thus, Conditions 2 and 3 are valid.
Calculating eAst and V(2||x£(T)||2), we obtain Wl\t) = I, W£12 (t) = e(l-e-t/£)I, W£21(t) = 0, W£22(t) = e-t/£I, v( l ||xe(T)||2) = x£(T). Therefore, equations (2.4) and (2.6) for l £ and l0 take the form
T
(1 — e~t/£ )2k lo
—l£ = x0 + e(1 — e~T/ )y° + J g ||) dt, —lo = x0 + TS\k) - (3^
0
If the vector-function fe(t) is such that fe(t) = 0(ea) as e — 0 for any a > 0 uniformly with respect to t £ [0; T] then instead of fe(t) we will write O. In particular, e-lT/e = O. From (3.2) we obtain
2
1. ||x0|| <T + 2=^ lo = - —- x0 and ||lo| < 2,
2 + T
x0 T
2. ||x0|| >T + 2=^ l0 = x0 and ||¿0M > 2-
x0
(3.3)
1. Consider first the case: ||x0|| <T + 2.
By virtue of (3.3) and Theorem 1 the inequality ||le|| < 2 is valid for any sufficiently small e. Taking into account that (1 — e-t/e) ^ 1 at any t ^ 0 and e > 0, from (3.2) we obtain for le the equation
T
-le = x0 + ey° + O + 2/(1 - e~t/e)2 dt le- (3.4)
T
Calculating the integral /(1 - e-t/e)2 dt = T - 3/(2e) + O, from (3.4) we find
0
= 4(x0 + ey0 + O)
le = - "
4 + 2T - 3e
It follows from this representation that le is expanded in the asymptotic series in powers of e.
Statement 3. Let ||x0|| < T + 2. Then the vector le which determines the optimal control in problem (3.1), is expanded as e — 0 in the power asymptotic series
le== l0 + ^ eklk, where, in particular, l1
4 + 2T k=1
2. Now consider the case: ||x0|| > T + 2.
By virtue of (3.3) and Theorem 1, the inequality ||le|| < 2 is valid for all sufficiently small e. Since for a fixed e the function (1 - e-t/e)||le|| increases monotonically from 0 at t = 0 into (1 - e-t/e)||le|| at t = T (which for sufficiently small e gives the inequality (1 - e-t/e)||le|| > 2), there is the unique t1>e £ (0;T) such that (1 - e-tl-s/e)HleH = 2, or
(1 - e-tm/e)||le|| =2, t1,e = -e ln(1 -j2j) ' (3.5)
Therefore, the equation (3.2) takes the form
tl,£ T
-le = x0 + e(1 - e-T/e)y0 + 2 J (1 - e-t/e)2 dtle + J (1 - e-t/e) dt. (3.6)
0 tM
Calculating the integrals in (3.6) and transposing (-le) into the right part, we obtain
1 1 1 , ( 2
0 = F(e, le) : = le + x0 + * - e-T/e)y0 - e (^ + ^ ^ 2 ln i1 - m)) l
KT+e ln i1 )+ee-T/e -e+e
(3.7)
Theorem 2. Let ||x0|| > T + 2. Then the vector l£ which determines the optimal control in problem (3.1) is expanded into a power asymptotic series (for e — 0)
l£ = Iq + ^ £kIk .
k=i
Proof. Consider the equation 0 = F(e,l), where F(■, ■) is defined in (3.7). Additionally predetermine e-T/£ at the point e = 0 as zero. Then we obtain that 0 = F(0,l0) and F(■, ■) is infinitely differentiable in e and l in a certain neighborhood of the point (0; l0). Since
F p :
dF(e, l)
dl
£=0,l = l
p = p +
p - (Iq, p)lo
T,
then operator F is continuously reversible and
F-'g = fg + T##) T+
(,3.8)
In this way, the theorem of implicitly specified function is applicable, which means that l£ (as a function of e) is infinitely differentiable in e for all small e and, therefore, l£ is expanded into the asymptotic series. The coefficients of this series can be found via the standard procedure: substituting the series into the equation (3.7), expanding values dependent on e into asymptotic series in power of e, and equaling terms of the same order of smallness with respect to e, we obtain an equation of the Flk = gk with the known right parts. Then, by the formula (3.8) we find lk.
In particular, for li we obtain the equation
Fli = gi := -£Q - y° +
1
+
1 1 , ,
— + 2 ln ( 1 -
2
)) lo -
ln 1
2
-1 +
2
lo
Hence, by virtue of (3.8) we obtain
li = gi +
T (lo,gi)lo\
T+
□
3
4. Remarks
1. Both in the first and the second cases under consideration, from (3.2), (3.5) and asymptotic expansion of l , the asymptotic expansions of both the quality index and optimal control as well as optimal state of the system are conventionally obtained. With this, the asymptotic expansions of the optimal control and optimal state of the system will be exponentially decreasing boundary layers in the neighborhood of point t = 0. Moreover, if t ^ e@ and ¡3 £ (0; 1), the optimal control uo(t) is a constant plus the asymptotic zero.
2. It follows from the formula (3.7) that l£ lies in the subspace n created by vectors x0 and y°. Therefore, for all t £ [0; T] and uO(t), x£ (t) and y£(t) lie in the same subspace n. In this way, the problem (3.1) is equivalent to the corresponding two-dimensional problem.
Acknowledgements
The author is very grateful to Prof. Alexey R. Danilin for the formulation of the problem and constant attention to the work.
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