CC BY
28
UDC 34F31, 34C23
On a correctness of boundary value problems for controllable systems with deviating argument1
© E. O. Burlakov, E. S. Zhukovskiy
Derzhavin Tambov State University, Tambov, Russia
In the paper, a continuous dependence on parameters of solutions to boundary value problems for functional-differential equations (including control functions) is studied. The results are then applied to a correctness of some boundary value problems and to the problem of continuous dependence on parameters of periodic solutions of controllable systems with deviating argument
Keywords: functional-differential equations, boundary value problems, periodic motions, continuous dependence on parameters, controllable systems with deviating argument
First let us introduce some spaces of functions on [a, b] with values in Rn, The space L([a, b], Rn) consists of summable (in sense of Lebesgue) functions y with the norm
IMU =/ |y(s)| ds,
J a
| ■ | being the Euclidean norm in Rn; the space L^([a,b], Rn) consists of measurable essentially bounded functions y with the norm ||y||L^ = vraisup |y(t)|, t E [a,b]; the space C([a,b], Rn) consists of continuous functions x with ||x||C = max |x(t)|, t E [a, b]; the space AC ([a, b], Rn) of absolutely continuous functions x is equipped bv the norm ||x||AC = |x(a)| + ||x||L.
Let E be a Banach space with the norm || ■ ||. By BE(u0, r) we denote the open
ball ||u — u0|| < r with center u0 and radius r, by A we denote the closure of A c E,
Let A be some Banach space. Consider the following boundary value problem:
j x = F(x, A); (1)
1 ^(x,A) = 0; ()
where F : AC ([a, b], Rn) x A ^ L([a, b], Rn), ^ : AC ([a, b], Rn) x A ^ Rm,
Suppose that for A0 E A problem (1) has a solution x0 E AC([a,b],Rn), Applying to (1) the implicit function theorem, we obtain:
1This work is supported by Russian Foundation for Basic Research (RFBR): grants 07-0100305 and 09-01-97503, Ministry Educ. Sci. RF: program DSPHS 2.1.1/1131, Norwegian National Program of Scientific Research FUGE attached to Norwegian Council of Scientific Research and Norwegian Committee for development of university science and education (NUFIJ): grant PRO 06/02.
1625
Theorem 11 Let: 1) there exist 50 > 0 a0 > 0 such that operators F, p are continuous and have continuous Frechet derivatives F^, in the neighbourhood
Bac(xo,ao) x £л(Ао,£о) °f (xo,Ao);
2) the operator L : AC ([a, b], Rn) ^ L([a, b], Rn) defined by Lz = z — F£ (x0, A0)z is surjective and dim Ker L = m;
3) the problem Lz = 0 tz = 0 has a unique solution z = 0, here t = pX(x0, A0), an operator AC([a,b], Rn) ^ Rm.
Then there exist 5 > 0 a > 0 such that for any A E Вл(A0,5) Йеге ercisfc a unique solution x = x(A) of problem (1) on Йе ball Bac(x0,a) and Йе 'map x(-) : Ba_(A0,5) ^ AC ([a, b], Rn) is continuous.
Remark 1. Let m = n, In this case the fulfilment of condition 2) follows, for example, from the Fredholm property of an operator Q : L([a, b], Rn) ^ L([a, b], Rn), a "principal part" of the operator L, defined by: Qy = Lz, where z is a primitive of y given bv z(t) = Ja y(s) ds.
Let M be ад open set in C([a, b],Rn) containing the solution x0 = x(A0) of problem (1) such that this solution is unique in M, Let Хл be the set of solutions of (1) corresponding to A and belonging to M,
Theorem 12 Let: 1) conditions of Theorem 1 are satisfied;
2) there exists 8 > 0 such that for each, A E Ba_(A0, 8) the opera tor F (■, A) can be extended to the space C([a,b], Rn);
3) for any r > 0 there exists a function gr in L([a,b], R) such that for all (x, A) in (Bc(x0,r)P| M) x Ba_(A0, 8) an estimate |(F(x,A))(t)| ^ gr(t) takes place almost
[a, b]
4) operator F(■, A0) is continuous on M.
5) for some sequence {A^} in the ball ВЛ(A0,8) convergent to A0 each set X^ contains at least two elements.
Then the set M is unbounded, and for each, i we can take xi in Хл. such that
IIx — x0||AC ^ 0, but for any оther xi E Хл., i. e. xi = xi; we have \\xi\\C ^ ro.
Let us apply these theorems to the following boundary value problem:
f x(t) = fx (HAx)(t),MA(t)) ,t E [a,b], (2)
[ x(a) — x(b) = ал;
where the superscript A ranges A, Here ал is a vector in Rn; a control ил is a function in L^([a, b], Rk ); by Hл we denote a linear op erator AC ([a, b], Rn) ^ L^([a, b], Rmn), so that Нлx = (H^x,..., H^x); a function fл defined on [a,b] x Rmn x Rk with values in Rn satisfies Karatheodori’ conditions:
Ki) function fл(-,y,u) is measurable for all y E Rmn, u E Rfc;
K2) function f л(£, ■, ■) is continuous for almost all t E [a, b] ;
K3) for any r > 0, there exists a function gr in L([a,b], R) such that for |y| ^ r and |u| ^ r one has ад estimate |fл(t,y,u)| ^ gr(t) for all A E Л and almost all t E [a, b]
1626
Let problem (2) has the solution x0 = x(A0) E AC ([a, b], Rn), Denote y0 = HA° x0.
Theorem 13 Let: 1) there exist a0 > 0 e0 > 0 50 > 0 and a fund ion G in L([a,b],R) such that for all A E Ba(A0, 50) and for almost all t E [a,b], all y E BRmn(y0(t), a0), a!I u E BRk(uA°(t),e0), there exist partial derivatives d/A/5yjP
t (y, u)
%
(t,y,u) ^ G(t), /,p = 1,...,n, j = 1,...,m;
2) operators H\ A E Ba(A0, 50), are uniformly bounded;
3) the problem Z(t) = A(t)(HA°z)(t), z(0) — z(w) = 0 (t E [a,b]) ftas a unique solution z = 0, here A(t) is an x nm-matrix ((5/A/dy) (t,y0(t), uA° (t))
Further, let A be an arbitrary element in Ba(A0,50) and X be a sequence convergimg to A (for simplicity we do not write an index of this sequence). We suppose
4) aA ^ aA;
5) for all x E AC([a,b], Rn), (HAx)(-) ^ (HAx)(-) in measure on [a, b];
6) uA(■) ^ uA(■) in measure on [a,b];
7) for any y E Rmn, /A(-, y, uA(■)) ^ /A(-, y, uA(-)) in measure on [a, b];
8) for any y E BLtc (y0,a0), if y(-) ^ y(-) in measure, then the 'matrix function
((d/A/dy) (■, X('), uA(-))j converges to the matrix function ((d/A/dy) (■, y(-), uA(-))) in measure.
Then there exist 5 > 0 ^ > 0 such that for any A E Ba(A0,5) ttere crisis a unique solution x = x(A) E AC([a,b], Rn) of problem (2) in the ball BAC (x0,a), and tte 'map x(-) : Ba(A0 ,5) ^ AC ([a, b], Rn) is continuous.
Introduce M, M, XA for problem (2) in the same way as above for problem (1),
Theorem 14 Let: 1) conditions of Theorem 3 are satisfied;
2) there exists 5 > 0, such that for all A E Ba (A0, 5) the opera tor HA can be extended to an operator on the space C([a, b], Rn);
3) for any r > 0 there exists r > 0 such that the inequality |(HAx)(t)| ^ r takes place for all (x, A) in (BC(x0, r) P| M) x Ba(A0, 5) and for almost all t E [a, b];
4) for all x E M, (HAx)(-) ^ (HAx)(-) in measure when A ^ A;
5) for some sequence {A^} in the ball Ba(A0,5) convergent to A0 each set XA. contains at least two elements.
Then the set M is unbounded, and for each, i we can take x in XAi such that ||x — x0||AC ^ 0, but for any other x^ E XA., i. e. x* = xi; we have |xi|C ^ ro.
Now we consider a continuous dependence of periodic solutions of controllable systems with deviating argument. For ordinary differential equations, such a question has been studied by E, L, Tonkov [1], [2].
1627
Introduce some spaces of w-periodic functions on R with values in Rn, The space Ln consists of summable functions y with the norm
pW
IlylU = |y(s)| ds;
0
the space L^ consists of measurable essentially bounded functions y with the norm IlylUrc = vraisup |y(t)U t E [0, w]; the space Cn of continuous func tions x is equipped bv the norm ||x||C = max |x(t)|, t E R; the space ACn of absolutely continuous functions x is equipped by the norm ||x||AC = |x(0)| + ||x||L,
Let us consider controllable systems:
x(t) = / (t,x(hi(t)),... ,x(hm(t)),u0(t)), (3)
x(t) = / (t, x(hM(t)),..., x(hm,i(t)), ui(t)). (3i)
Here t E R, i = 1, 2,..., functions hj, j : R — R (j = 1,..., m) are measurable and w-periodic, functions u0,u : R — Rk are measurable essentially bounded and w-periodic, a function / : R x Rn x ... x Rn x Rk — Rn is w-periodic on the first argument and satisfies Karatheodori’ conditions.
Let equation (3) has an w-periodic solution x0 E ACn, Denote y0 = H0x0 =
(x(hi(-)),...,x(hm(-)) e L^n,
Theorem 15 Let: 1) there exist 50 > 0 a0 > 0 such that for almost all t E [0, w],
all y E BRmn(y0(t),a0), a/1 u E BRk(uA°(t), 50) partial derivatives 5/l/5yj-p(t, y, u) (cf Theorem 3) exist and satisfy Karatheodori’ conditions;
2) when i — ro, sequences {u^} u {h^} converge in measure to u0 and hj, respectively;
3) the equation Z(t) = A(t)(H0z)(t), where A(t) is the matrix as in Theorem 3 (without A) has a unique w-periodic solution z = 0.
Then there exist a > 0 and a numher I such that for all i > I there exist a unique w-periodic solution x^ E ACn of equation (3i) satisfying the inequality ||xi — x01|ac < a, and |x^ — x0||ac —— 0.
Theorem 16 Let M be an open set in Cn containing the solution x0 of problem (3) such that this solution is unique in M. Then, if each (3i) has more than one w-periodic solution in M, then M is unbounded and one can take a sequence x^ of w-periodic solutions of (3i) such that ||xj||C — ro when i — ro.
References
1, E, L, Tonkov. Optimal periodic motions of controllable systems, Math, Physics, 1977, vol. 21, 45-59.
2, E. L. Tonkov. Optimal control on periodical motions, Math. Physics, 1977, vol. 22, 54-64.
1628
