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The controllability of boundary-value problems for quasilinear impulsive systems
Nonlinear Analysis 34 (1998) 1055 – 1065
The controllability of boundary-value problems for quasilinear impulsive systems M.U. Akhmetov 1 , A. Zafer ∗ Department of Mathematics, Middle East Technical University, 06531 Ankara, Turkey Received 3 March 1997; accepted 14 July 1997
Keywords: Impulse; Control; Boundary-value problem; Quasilinear system
1. Introduction The question of the control of linear and quasilinear systems of ordinary di erential equations has attracted the attention of many authors for several years [3, 4, 7, 9]. Recently, there has been a lot of activity in the investigation of di erential equations with impulse e ect [1, 2, 6, 8, 10]. The controllability of boundary-value problems for linear impulsive systems was considered in [2]. In this paper, by the help of some results from [1, 2, 6, 8, 10], we will investigate the problem of the control of boundary-value problems for quasilinear impulsive systems. We will consider not only xed but also variable moments of impulse control. A comparison method is used to investigate the systems with variable moments of impulse actions. Let and be xed real numbers such that ¡ , and r and p be xed positive integers. Denote by Lr2 [ ; ] the set of all square integrable and bounded functions : [ ; ] → R r and by D r [1; p] the set of all nite sequences {i }, i ∈ R r , i = 1; : : : ; p.
∗
Corresponding author. E-mail: [email protected].
1
Present address: Higher Educational Institution “Dunie”, 10a, 101 Strelkovoy Brigady, Aktyubinsk 463000, Kazakhstan. The author is thankful to The Scienti c and Technical Research Council of Turkey ˙ITAK) for the nancial support. (TUB 0362-546X/98/$19.00 ? 1998 Elsevier Science Ltd. All rights reserved. PII: S 0 3 6 2 - 5 4 6 X ( 9 8 ) 0 0 0 0 5 - 4
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M.U. Akhmetov, A. Zafer / Nonlinear Analysis 34 (1998) 1055 – 1065
We de ne a space rp = Lr2 × D r and denote its elements by {; }, and let Z h{; }; {w; }i =
(; w) dt +
p X
(i ; i )
i=1
be an inner product in rp , where ( ; ) is the euclidean scalar product in R r : The main aim of this paper is to consider the problem of the control of systems of di erential equations with impulse actions on surfaces. The system under consideration is of the form dx = A(t)x(t) + C(t)u + f(t) + g(t; x; u; ); dt x|t=i + i (x; ) = Bi x + Di vi + Ji + Wi (x; vi ; )
t 6= i + i (x; ); (1)
with the boundary condition x( ) = a;
x( ) = b;
(2)
where x ∈ Rn , the symbol x|t= means x(+) − x(), A and C are matrix functions of the sizes (n × n) and (n × m), respectively, the elements of which belong to L12 [ ; ], m ≤ n, {i }, i = 1; : : : ; p, is a strictly increasing sequence of real numbers in ( ; ), Bi and Di are, respectively, (n × n) and (n × m) constant matrices with det(I + Bi ) 6= 0, i = 1; : : : ; p, {f; J } ∈ np [ ; ], is a small positive parameter, g, Wi , and i are continuous and continuously di erentiable functions in x, u, and v. The control problem (1) and (2), which we shall denote by , is said to be solvable if given any bounded set G ⊂ Rn there exists a positive 0 ∈ R; 0 = 0 (G), such that for all arbitrary a; b ∈ G and ¡0 there is a control {u; v} ∈ pm for which system (1) admits a solution x(t) satisfying Eq. (2). The process de ned by Eq. (1) for xed and {u; v} operates as follows: the point Pt (t; x(t)), starting at (t0 ; x0 ), moves along the curve de ned by the solution x(t) = x(t; t0 ; x0 ) of the equation dx = A(t)x(t) + C(t)u + f(t) + g(t; x; u; ): dt
(3)
The motion along this curve terminates at time t = i when the point Pt arrives at one of the surfaces of discontinuity so that i = i + i (x(i ); ): At that moment the point Pt performs a jump x|t=i = Bi x(i ) + Di vi + Ji + Wi (x(i ); vi ; ) and proceeds to move along the curve described by the solution x(t; i ; x(i +)) of system (3), until it meets another surface of discontinuity, and so on. We should note that system (1) considered in this paper belongs to a class of systems with impulses at non- xed moments. It is possible for the integral curve of this system to meet more than one time and even in nitely many times one and the same surface of discontinuity. This phenomenon is called beating [8, 10]. Therefore, the investigation of such a system needs conditions for the absence of beating. Below we shall rst deal with this consideration.
M.U. Akhmetov, A. Zafer / Nonlinear Analysis 34 (1998) 1055 – 1065
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Let s be a positive real number, and let s be the subspace of elements (x; u; v) satisfying the inequality |x| + |u| + |v| ≤ s, where | · | is the euclidean norm in Rn , and let Gs = {(x; u; v; t; i; )|(x; u; v) ∈ s ; ≤ t ≤ ; i = 1; : : : ; p; ≤ 1 }; where 1 is a xed positive real number. Fix a set G ∈ Rn , and a positive real number h ∈ R such that for all x ∈ G, |x|¡h: Let H ¿h be a real number, and set m1 = max sup |A(t)|; sup |C(t)|; max |Bi | ;
t
t
i
m2 = max sup |f(t)|; max |Ji | i
t
and
m3 = max max |g|; max |W |; max | | : GH
GH
GH
If it is assumed that 1 m3 ¡ min{1 − ; − p } and that for any x, ; and i from GH , the relation i+1 + i+1 (x; )¡i + i (x; ) is true, then it follows that every solution of Eq. (1), which is in GH and de ned on [ ; ], intersects each of the surface t = i + i (x; ), i = 1; : : : ; p. In view of the di erentiability of functions g, Wi , and i , there exists a positive real number l such that uniformly in GH , |g(t; x1 ; u1 ; v1 ; ) − g(t; x2 ; u2 ; v2 ; )| ≤ l{|x1 − x2 | + |u1 − u2 | + |v1 − v2 |}; |Wi (x1 ; v1 ; ) − Wi (x2 ; v2 ; )| ≤ l{|x1 − x2 | + |v1 − v2 |}; | i (x1 ; ) − i (x2 ; )| ≤ l |x1 − x2 |: Now if we let 2 ¡1 be a positive real number which satis es 2 l(2m1 H + m2 + 2 m3 )¡1; then we can obtain the following lemma. The proof is similar to that of Lemma 5 on p. 22 in [10], and hence is omitted. Lemma 1. Let system (1) be considered in GH ; and let ¡2 . If the relation i (x; ) ≥ i (x + Bi x + Di v + Ji + Wi (x; v; ); ) is valid; then every solution x(t) of Eq. (1) meets any given surface t = i + i (x; ) at most once. To investigate system (1) we shall use a comparison method [6]. Let x0 (t) be a solution of system (3) with initial condition x0 (i ) = x for a xed i, i = 1; : : : ; p, and
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M.U. Akhmetov, A. Zafer / Nonlinear Analysis 34 (1998) 1055 – 1065
t = i be the instant of meet of solution x0 (t) with the surface t = i + i (x; ). Suppose that x1 (t) is a solution of the Cauchy problem x1 (i ) = (I + Bi )x0 (i ) + Di vi + Ji + Wi (x0 (i ); vi ; ) for system (3). We de ne Z Si (x; u; v; ) = (I + Bi )
i
i
[A(t)x0 (t) + C(t)u(t) + f(t) + g(t; x0 (t); u(t); )] dt Z
+Wi (x0 (i ); vi ; ) +
i
i
[A(t)x1 (t) + C(t)u(t) + f(t)
+g(t; x1 (t); u(t); )] dt:
(4)
As in [6] one can easily show that each Si , i = 1; : : : ; p; is a continuously di erentialable function in x and vi . Using the de nition of Si , we see that system (1) and dy = A(t)y + C(t)u + f(t) + g(t; y; u; ); dt y|t=i = Bi y + Di vi + Ji + Si (y; u; vi ; )
t 6= i ; (5)
have the property in GH [6]. This means that if ¡2 , where 2 satis es 2 m2 (2m1 H +m2 +2 m3 )¡H −h, then given any solution x(t) of (1), |x(t)|¡h, t ∈ [ ; ], there is a solution y(t) of (5), |y(t)|¡H , such that x(t) = y(t) for all t ∈ [ ; ] except possibly at points t ∈ [i ; i ], i = 1; : : : ; p. Conversely, given any solution y(t) of Eq. (5), |y(t)|¡h, t ∈ [ ; ], there is a solution x(t) of Eq. (1), |x(t)|¡H , such that x(t) = y(t) for all t ∈ [ ; ] except possibly at points t ∈ [i ; i ], i = 1; : : : ; p. ˆ be Now, we look at the dependence of Si on the control function u(t). So let u(t) another control function, and let y0 (t); y0 (i ) = x, be a solution of the system dx = A(t)x(t) + C(t)u(t) ˆ + f(t) + g(t; x; u(t); ˆ ): (6) dt Suppose that t = i is the instant of meet of y0 (t) with the surface t = i + i (x; ): Without loss of generality, we can assume that i ¡i ¡i . Let y1 (t) be also a solution of Eq. (6) satisfying the initial condition y1 (i ) = (I + Bi )y0 (i ) + Di vi + Ji + Wi (y0 (i ); vi ; ): Clearly,
Z
Si (x; u; ˆ v; ) = (I + Bi )
i
i
[A(t)y0 (t) + C(t)u(t) ˆ + f(t) + g(t; y0 (t); u(t); ˆ )] dt
+Wi (y0 (i ); vi ; ) +
Z
i
i
[A(t)y1 (t) + C(t)u(t) ˆ + f(t)
ˆ )] dt: +g(t; y1 (t); u(t);
(7)
Pick a positive real number 3 ≤ 2 such that 3 l(2m1 H + 3 m3 + m2 )¡1
(8)
M.U. Akhmetov, A. Zafer / Nonlinear Analysis 34 (1998) 1055 – 1065
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and de ne ku − uk ˆ 1 = max |u − u|: ˆ i ≤t≤i
The following lemma holds. Lemma 2. Suppose that (x; u(t); v) and (x; u(t); ˆ v) for all t ∈ [i ; i ] are in h . Then |Si (x; u; v; ) − Si (x; u; ˆ v; )| ≤ k(H )ku − uk ˆ 1
(9)
for all ¡3 and t ∈ [i ; i ]; where k(H ) is a bounded function. Proof. It is not dicult to verify that the solutions x0 ; x1 of Eq. (3) and the solutions y0 ; y1 of Eq. (6) which we have used in Eqs. (4) and (7) remain in GH . Subtracting Eq. (4) from Eq. (7) we nd that Z i ˆ v; ) − Si (x; u; v; ) = {(I + Bi )A(t)(y0 (t) − x0 (t)) − A(t)(y1 (t) − x1 (t)) Si (x; u; i
ˆ ) − g(t; x0 (t); u(t); )) +[(I + Bi )(g(t; y0 (t); u(t); ˆ ) − g(t; x1 (t); u(t); ))]} dt −(g(t; y1 (t); u(t); Z i +Bi C(t)(u(t) ˆ − u(t)) dt i
+[Wi (y0 (i ); vi ; ) − Wi (x0 (i ); vi ; )] Z i − {(I + Bi )A(t)x0 (t) − A(t)x1 (t) + Bi C(t)u(t) i
+[(I + Bi )g(t; x0 (t); u(t); ) − g(t; x1 (t); u(t); )]} dt: (10) Since the right-hand side of Eq. (3) satis es a Lipschitz condition in u, there exists a constant l1 , see [5], such that |y0 (t) − x0 (t)| ≤ l1 ku − uk ˆ 1:
(11)
It follows that |g(t; y0 (t); u(t); ˆ ) − g(t; x0 (t); u(t); )| ≤ l(1 + l1 )ku − uk ˆ 1: On the other hand, since Eq. (8) is satis ed and i − i = i (x0 (i ); ) − i (y0 (i ); ) Z i [A(t)x0 (t) + C(t)u(t) + f(t) + g(t; x0 (t); u(t); )] dt ≤ l i ! ˆ 1 +l1 ku − uk
ˆ 1 ]; ≤ l[(i − i )(2m1 H + m3 + m2 ) + l1 ku − uk
(12)
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M.U. Akhmetov, A. Zafer / Nonlinear Analysis 34 (1998) 1055 – 1065
we have i − i ≤
ll1 ku − uk ˆ 1; 1 − lh1 (H; )
h1 (H; ) = 2m1 H + m3 + m2 :
(13)
Using Eq. (13) and the expression Z i [A(t)x1 (t) + C(t)u(t) + f(t) + g(t; x1 (t); u(t); )] dt; x1 (i ) = x1 (i ) + i
we nd that ˆ 1; |x1 (i ) − x1 (i )| ≤ h2 (H; )ku − uk
h2 (H; ) =
ll1 h1 (H; ) : 1 − lh1 (H; )
It follows that |y1 (t) − x1 (t)| ≤ l2 h2 (H; )ku − uk ˆ 1;
(14)
where l2 is a xed positive real number. Now, we shall estimate the integrals in Eq. (10). Denote them in order by I1 , I2 , and I3 . Using Eqs. (11), (12), and (14) we obtain |I1 | ≤ m3 {(1 + m1 )m1 l1 + m1 h2 (H; ) + [(1 + m1 )l(1 + l1 ) ˆ 1 = k 1 (H )ku − uk ˆ 1; +l(1 + h2 (H; ))]}ku − uk |I2 | ≤ m21 m3 ku − uk ˆ 1 = k 2 (H )ku − uk ˆ 1
(15) (16)
and |I3 | ≤
ll1 {(1 + m1 )m2 H + (m1 + m21 )H 1 − lh2 (H; ) ˆ 1: +[(1 + m1 )m3 + m3 ]} = k 3 (H )ku − uk
(17)
Moreover, |Wi (y0 (i ); vi ; ) − Wi (x0 (i ); vi ; )| ≤ l(l1 + h2 (H; ))ku − uk ˆ 1 = k 4 (H )ku − uk ˆ 1:
(18)
By summing relations (15)–(18) and noting that k i , i = 1; : : : ; 4, are bounded functions of H , we see that Eq. (9) holds. Thus, the lemma is proved. In what follows, we denote by X (t); X ( ) = I , a fundamental matrix of dx = A(t)x; t 6= i dt x|t=i = Bi x
M.U. Akhmetov, A. Zafer / Nonlinear Analysis 34 (1998) 1055 – 1065
and de ne (t) =
Z
t
Q(t)QT (t) dt +
X
1061
Pi PiT ;
¡i ¡t
where Q(t) = X −1 (t)C(t) and Pi = X −1 (i )Di . The problem when = 0 was considered in [2] and the following theorems were obtained. Theorem 1. The problem 0 is solvable if and only if the matrix ( ) is non-singular. Theorem 2. Let 0 be solvable. Then the pair {u; v} ∈ pm is a solving control for 0 if and only if it has the form ˆ u = QT (t)c + u(t); where
t ∈ [ ; ];
vi = PiT c + vˆi ;
"
c = −1 ( ) X −1 ( )b − X −1 ( )a −
Z
i = 1; : : : ; p;
X −1 (t)f(t) dt −
p X
# X −1 (i )Ji ;
i=1
and {u; ˆ v} ˆ ∈ pm is orthogonal to all columns of [QT ; PiT ]. 2. Control of systems with xed moments of impulse actions Consider the following system: dx = A(t)x(t) + C(t)u + f(t) + g(t; x; u; ); dt
t 6= i
x|t=i = Bi x + Di vi + Ji + Wi (x; vi ; )
(19)
with boundary condition (2). All quantities here are as in Eq. (1) except that the moments of impulse actions are now xed. We denote this problem by . Theorem 3. If the matrix ( ) is non-singular; then is solvable. The solving control of this problem is the limit of a uniformly convergent sequence obtained by the method of successive approximations. Proof. We will prove that is solvable with a control {u; v} of the following form: ˆ u = QT (t)c + u(t);
t ∈ [ ; ];
vi = PiT c + vˆi ;
i = 1; : : : ; p;
(20)
ˆ v} ˆ ∈ pm is orthogonal to all columns of where c ∈ Rn is a constant vector and {u; T T [Q ; Pi ].
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M.U. Akhmetov, A. Zafer / Nonlinear Analysis 34 (1998) 1055 – 1065
The problem is equivalent to solving Z t Q(s)(u(s) + C −1 (s)f(s) + C −1 (s)g(s; x(s); u(s); )) ds x(t) = X (t) a +
X
+
Pi (vi + Di−1 Ji + Di−1 Wi (x(i ); vi ; )) ;
x( ) = b:
(21)
¡i ¡t
Substituting Eq. (20) into Eq. (21) we obtain the vector c as " Z −1 −1 −1 X −1 (t)(f(t) + g(t; x(t); u(t); )) dt c = ( ) X ( )b − X ( )a −
p
−
X
#
X −1 (i ) (Ji + Wi (x(i ); vi ; )) :
(22)
i=1
If we let u0 (t) = QT (t) ( )−1 K + u(t); ˆ vi0 = PiT ( )−1 K + vˆi ; Z t X (Q(s)u0 (s) + X −1 (s)f(s)) ds + (Pi vi0 + X −1 (i )Ji ) ; x0 (t) = X (t) a +
¡i ¡t
0 = (x0 (t); u0 (t); vi0 ); = (x(t); u(t); vi ); Z t X −1 (s)g(s; x(s); u(s); ) ds; (t; ; ) =
(t; ; ) =
X
X −1 (i )Wi (x(i ); vi ; );
¡i ¡t
where K = X −1 ( )b − X −1 ( )a −
Z
X −1 (t)f(t) dt −
p X
X −1 (i )Ji ;
i=1
then, by substituting Eq. (22) into Eq. (21) we nd that satis es the following relation: = 0 + P( ; ); where P = (P1 ; P2 ; Pi ), P1 (t; ) = X (t)[(t; ; ) + (t; ; ) − (t) −1 ( )(( ; ; ) + ( ; ; ))]; P2 (t; ) = Q(t)T −1 ( )[( ; ; ) + ( ; ; )]; Pi ( ) = PiT −1 ( )[( ; ; ) + ( ; ; )]:
(23)
M.U. Akhmetov, A. Zafer / Nonlinear Analysis 34 (1998) 1055 – 1065
1063
We introduce the norm k · k, k k = max |x(t)| + max |u(t)| + max |vi |; t
t
i
in the space of all elements of the form = (x(t); u(t); vi ). Suppose that the number h which is xed earlier satis es h¿k 0 k; and that 4 is a positive real number such that 4 ≤ 1 and 4 max kP( ; )k¡h − k 0 k: k k≤h 0¡≤1
Let be the subspace consisting of elements = (x(t); u(t); vi ) such that k k ≤ h, where x(t) is piecewise absolutely continuous, continuous on the left, and has discontinuities of the rst kind at moments i ; i = 1; : : : ; p; and {u; v} ∈ pm . It follows that if ≤ 4 then the operator 0 + P( ; ) maps to itself. We shall show that if is suciently small then P is a contraction mapping. Let m4 = max max |Q(t)T −1 ( )X −1 (s)|; max |PiT −1 ( )X −1 (s)|; t; s
max |X (t)X t; s
i; s
−1
−1
(s)|; max |X (t) (t) t; s
( )X
−1
(s)| :
Then, we have kP( 1 ; ) − P( 2 ; )k ≤ 4m4 l( − + p)k 1 − 2 k uniformly with respect to t ∈ [ ; ], ∈ [0; 4 ], and i = 1; : : : ; p for any 1 ; 2 ∈. Now if we let 0 ¡4 be such that 40 m4 l( − + p)¡1; then for ¡0 , P becomes a contraction mapping. Thus, we can obtain a sequence { i }, i = 0; 1; 2; : : : ; i+1 = 0 +P( i ; ) converging to some element 0 ∈ , 0 = (x0 ; u0 ; v0 ). Substituting 0 in Eq. (19), di erentiating x0 (t), and checking the boundary condition (2) and the condition of discontinuities, one can verify that 0 solves . Thus, the proof is complete. 3. Control of systems with variable moments of impulse actions Now, we return to considering problem . In view of the property of systems (1) and (5), the problem is reduced to a problem of the kind for system (5). Let us denote it by . The following lemma is concerned with the solvability of .
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M.U. Akhmetov, A. Zafer / Nonlinear Analysis 34 (1998) 1055 – 1065
Lemma 3. If ( ) is non-singular then is solvable. Proof. Let ¡3 , where 3 is as in Eq. (8), and let (x; u; v) ∈ h . Then using Eq. (4) we obtain |Si (x; u; vi ; )| ≤m5 ; where m5 = m3 [(2 + m1 )(2m1 h + m2 + 3 m3 ) + 3 m3 ]. Since Si is di erentiable in x and vi , there is a positive real number 5 such that for ¡5 it satis es a Lipschitz condition in x and v. The corresponding Lipschitz constants have the form k2 (H ) and k3 (H ), where k2 and k3 are bounded functions. Setting k(H ) = max{k1 ; k2 ; k3 } and 6 = min{3 ; 5 }, we see, in view of Lemma 2, that if ¡6 then |Si (x1 ; u1 ; v1 ; ) − Si (x2 ; u2 ; v2 ; )| ≤ k(H ){|x1 − x2 | + ku1 − u2 k1 + |v1 − v2 |} uniformly in h . Now, we let 0 ∈ be as in Theorem 3 and consider the operator = 0 +T( ; ) in , where T = (T1 ; T2 ; Ti ), T1 ( ; ) = X (t)[(t; ; ) + ˜ (t; ; ) − (t) −1 ( )(( ; ; ) + ˜ ( ; ; ))]; T2 ( ; ) = Q(t)T −1 ( )[( ; ; ) + ˜ ( ; ; )]; Ti ( ; ) = PiT −1 ( )[( ; ; ) + ˜ ( ; ; )] with (t; ; ) as de ned in Eq. (23) and X ˜ (t; ; ) = 1 X −1 (i )Si (x(i ); u; vi ; ): ¡i ¡t
Choosing 0 as 0 = min 6 ;
4(h − k 0 k) 4 ; ( − )m3 m4 + pm4 m5 lm4 ( − ) + k(H )m4p
;
we can see similarly as in the proof of Theorem 3 that if ¡0 , the operator 0 + T( ; ) maps to itself and is contractive. Thus, the lemma is proved. Theorem 4. If () is non-singular then is solvable. Proof. Lemma 3 implies that there is a control {u; ˆ v} ˆ such that system (5) has a solution y(t) which satis es y( ) = a and y( ) = b. Since Eqs. (1) and (5) have the property , system (1) with the control {u; ˆ v} ˆ admits a solution x(t) such that x(t) = y(t) for all t except for t ∈ [i ; i ], i = 1; : : : ; p. Since and are not in [i ; i ], i = 1; : : : ; p, the solution x(t) also satis es the boundary condition (2). This completes the proof.
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References [1] M.U. Akhmetov, N.A. Perestyuk, On comparison method for di erential systems with pulse e ect, Di erential Equations 9 (26) (1990) 1079 –1086. [2] M.U. Akhmetov, N.A. Perestyuk, M.A. Tleubergenova, The control of linear impulsive systems, Ukrainian. Math. Zh. 3 (47) (1995) 307–314. [3] S.A. Aysagaliev, K.O. Onaybaer, T.G. Mazakov, The controllability of nonlinear systems, Izv. Akad. Nauk. Kazakh.-SSR.-Ser. Fiz.-Mat (1) (1985) 83 –87. [4] S. Barnett, Introduction to Mathematical Control Theory, Clarendon Press, Oxford, 1975. [5] P. Hartman, Ordinary Di erential Equations, Wiley, New York, 1964. [6] A. Halanay, D. Wexler, Qualitative Theory of Impulse Systems, Editura Academiei Republici Socialiste Romania, Bucuresti, 1968. [7] N.N. Krasovsky, Theory of Motion Control, Linear Systems, Nauka, Moscow, 1973. [8] V. Lakshmikantham, D.D. Bainov, P.S. Simeonov, Theory of Impulsive Di erential Equations, World Scienti c, Singapore, 1989. [9] E.B. Lee, L. Markus, Foundations of Optimal Control Theory, Wiley, New York, 1967. [10] A.M. Samoilenko, N.A. Perestyuk, Impulsive Di erential Equations, World Scienti c, Singapore, 1995.
The controllability of boundary-value problems for quasilinear impulsive systems M.U. Akhmetov 1 , A. Zafer ∗ Department of Mathematics, Middle East Technical University, 06531 Ankara, Turkey Received 3 March 1997; accepted 14 July 1997
Keywords: Impulse; Control; Boundary-value problem; Quasilinear system
1. Introduction The question of the control of linear and quasilinear systems of ordinary di erential equations has attracted the attention of many authors for several years [3, 4, 7, 9]. Recently, there has been a lot of activity in the investigation of di erential equations with impulse e ect [1, 2, 6, 8, 10]. The controllability of boundary-value problems for linear impulsive systems was considered in [2]. In this paper, by the help of some results from [1, 2, 6, 8, 10], we will investigate the problem of the control of boundary-value problems for quasilinear impulsive systems. We will consider not only xed but also variable moments of impulse control. A comparison method is used to investigate the systems with variable moments of impulse actions. Let and be xed real numbers such that ¡ , and r and p be xed positive integers. Denote by Lr2 [ ; ] the set of all square integrable and bounded functions : [ ; ] → R r and by D r [1; p] the set of all nite sequences {i }, i ∈ R r , i = 1; : : : ; p.
∗
Corresponding author. E-mail: [email protected].
1
Present address: Higher Educational Institution “Dunie”, 10a, 101 Strelkovoy Brigady, Aktyubinsk 463000, Kazakhstan. The author is thankful to The Scienti c and Technical Research Council of Turkey ˙ITAK) for the nancial support. (TUB 0362-546X/98/$19.00 ? 1998 Elsevier Science Ltd. All rights reserved. PII: S 0 3 6 2 - 5 4 6 X ( 9 8 ) 0 0 0 0 5 - 4
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M.U. Akhmetov, A. Zafer / Nonlinear Analysis 34 (1998) 1055 – 1065
We de ne a space rp = Lr2 × D r and denote its elements by {; }, and let Z h{; }; {w; }i =
(; w) dt +
p X
(i ; i )
i=1
be an inner product in rp , where ( ; ) is the euclidean scalar product in R r : The main aim of this paper is to consider the problem of the control of systems of di erential equations with impulse actions on surfaces. The system under consideration is of the form dx = A(t)x(t) + C(t)u + f(t) + g(t; x; u; ); dt x|t=i + i (x; ) = Bi x + Di vi + Ji + Wi (x; vi ; )
t 6= i + i (x; ); (1)
with the boundary condition x( ) = a;
x( ) = b;
(2)
where x ∈ Rn , the symbol x|t= means x(+) − x(), A and C are matrix functions of the sizes (n × n) and (n × m), respectively, the elements of which belong to L12 [ ; ], m ≤ n, {i }, i = 1; : : : ; p, is a strictly increasing sequence of real numbers in ( ; ), Bi and Di are, respectively, (n × n) and (n × m) constant matrices with det(I + Bi ) 6= 0, i = 1; : : : ; p, {f; J } ∈ np [ ; ], is a small positive parameter, g, Wi , and i are continuous and continuously di erentiable functions in x, u, and v. The control problem (1) and (2), which we shall denote by , is said to be solvable if given any bounded set G ⊂ Rn there exists a positive 0 ∈ R; 0 = 0 (G), such that for all arbitrary a; b ∈ G and ¡0 there is a control {u; v} ∈ pm for which system (1) admits a solution x(t) satisfying Eq. (2). The process de ned by Eq. (1) for xed and {u; v} operates as follows: the point Pt (t; x(t)), starting at (t0 ; x0 ), moves along the curve de ned by the solution x(t) = x(t; t0 ; x0 ) of the equation dx = A(t)x(t) + C(t)u + f(t) + g(t; x; u; ): dt
(3)
The motion along this curve terminates at time t = i when the point Pt arrives at one of the surfaces of discontinuity so that i = i + i (x(i ); ): At that moment the point Pt performs a jump x|t=i = Bi x(i ) + Di vi + Ji + Wi (x(i ); vi ; ) and proceeds to move along the curve described by the solution x(t; i ; x(i +)) of system (3), until it meets another surface of discontinuity, and so on. We should note that system (1) considered in this paper belongs to a class of systems with impulses at non- xed moments. It is possible for the integral curve of this system to meet more than one time and even in nitely many times one and the same surface of discontinuity. This phenomenon is called beating [8, 10]. Therefore, the investigation of such a system needs conditions for the absence of beating. Below we shall rst deal with this consideration.
M.U. Akhmetov, A. Zafer / Nonlinear Analysis 34 (1998) 1055 – 1065
1057
Let s be a positive real number, and let s be the subspace of elements (x; u; v) satisfying the inequality |x| + |u| + |v| ≤ s, where | · | is the euclidean norm in Rn , and let Gs = {(x; u; v; t; i; )|(x; u; v) ∈ s ; ≤ t ≤ ; i = 1; : : : ; p; ≤ 1 }; where 1 is a xed positive real number. Fix a set G ∈ Rn , and a positive real number h ∈ R such that for all x ∈ G, |x|¡h: Let H ¿h be a real number, and set m1 = max sup |A(t)|; sup |C(t)|; max |Bi | ;
t
t
i
m2 = max sup |f(t)|; max |Ji | i
t
and
m3 = max max |g|; max |W |; max | | : GH
GH
GH
If it is assumed that 1 m3 ¡ min{1 − ; − p } and that for any x, ; and i from GH , the relation i+1 + i+1 (x; )¡i + i (x; ) is true, then it follows that every solution of Eq. (1), which is in GH and de ned on [ ; ], intersects each of the surface t = i + i (x; ), i = 1; : : : ; p. In view of the di erentiability of functions g, Wi , and i , there exists a positive real number l such that uniformly in GH , |g(t; x1 ; u1 ; v1 ; ) − g(t; x2 ; u2 ; v2 ; )| ≤ l{|x1 − x2 | + |u1 − u2 | + |v1 − v2 |}; |Wi (x1 ; v1 ; ) − Wi (x2 ; v2 ; )| ≤ l{|x1 − x2 | + |v1 − v2 |}; | i (x1 ; ) − i (x2 ; )| ≤ l |x1 − x2 |: Now if we let 2 ¡1 be a positive real number which satis es 2 l(2m1 H + m2 + 2 m3 )¡1; then we can obtain the following lemma. The proof is similar to that of Lemma 5 on p. 22 in [10], and hence is omitted. Lemma 1. Let system (1) be considered in GH ; and let ¡2 . If the relation i (x; ) ≥ i (x + Bi x + Di v + Ji + Wi (x; v; ); ) is valid; then every solution x(t) of Eq. (1) meets any given surface t = i + i (x; ) at most once. To investigate system (1) we shall use a comparison method [6]. Let x0 (t) be a solution of system (3) with initial condition x0 (i ) = x for a xed i, i = 1; : : : ; p, and
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M.U. Akhmetov, A. Zafer / Nonlinear Analysis 34 (1998) 1055 – 1065
t = i be the instant of meet of solution x0 (t) with the surface t = i + i (x; ). Suppose that x1 (t) is a solution of the Cauchy problem x1 (i ) = (I + Bi )x0 (i ) + Di vi + Ji + Wi (x0 (i ); vi ; ) for system (3). We de ne Z Si (x; u; v; ) = (I + Bi )
i
i
[A(t)x0 (t) + C(t)u(t) + f(t) + g(t; x0 (t); u(t); )] dt Z
+Wi (x0 (i ); vi ; ) +
i
i
[A(t)x1 (t) + C(t)u(t) + f(t)
+g(t; x1 (t); u(t); )] dt:
(4)
As in [6] one can easily show that each Si , i = 1; : : : ; p; is a continuously di erentialable function in x and vi . Using the de nition of Si , we see that system (1) and dy = A(t)y + C(t)u + f(t) + g(t; y; u; ); dt y|t=i = Bi y + Di vi + Ji + Si (y; u; vi ; )
t 6= i ; (5)
have the property in GH [6]. This means that if ¡2 , where 2 satis es 2 m2 (2m1 H +m2 +2 m3 )¡H −h, then given any solution x(t) of (1), |x(t)|¡h, t ∈ [ ; ], there is a solution y(t) of (5), |y(t)|¡H , such that x(t) = y(t) for all t ∈ [ ; ] except possibly at points t ∈ [i ; i ], i = 1; : : : ; p. Conversely, given any solution y(t) of Eq. (5), |y(t)|¡h, t ∈ [ ; ], there is a solution x(t) of Eq. (1), |x(t)|¡H , such that x(t) = y(t) for all t ∈ [ ; ] except possibly at points t ∈ [i ; i ], i = 1; : : : ; p. ˆ be Now, we look at the dependence of Si on the control function u(t). So let u(t) another control function, and let y0 (t); y0 (i ) = x, be a solution of the system dx = A(t)x(t) + C(t)u(t) ˆ + f(t) + g(t; x; u(t); ˆ ): (6) dt Suppose that t = i is the instant of meet of y0 (t) with the surface t = i + i (x; ): Without loss of generality, we can assume that i ¡i ¡i . Let y1 (t) be also a solution of Eq. (6) satisfying the initial condition y1 (i ) = (I + Bi )y0 (i ) + Di vi + Ji + Wi (y0 (i ); vi ; ): Clearly,
Z
Si (x; u; ˆ v; ) = (I + Bi )
i
i
[A(t)y0 (t) + C(t)u(t) ˆ + f(t) + g(t; y0 (t); u(t); ˆ )] dt
+Wi (y0 (i ); vi ; ) +
Z
i
i
[A(t)y1 (t) + C(t)u(t) ˆ + f(t)
ˆ )] dt: +g(t; y1 (t); u(t);
(7)
Pick a positive real number 3 ≤ 2 such that 3 l(2m1 H + 3 m3 + m2 )¡1
(8)
M.U. Akhmetov, A. Zafer / Nonlinear Analysis 34 (1998) 1055 – 1065
1059
and de ne ku − uk ˆ 1 = max |u − u|: ˆ i ≤t≤i
The following lemma holds. Lemma 2. Suppose that (x; u(t); v) and (x; u(t); ˆ v) for all t ∈ [i ; i ] are in h . Then |Si (x; u; v; ) − Si (x; u; ˆ v; )| ≤ k(H )ku − uk ˆ 1
(9)
for all ¡3 and t ∈ [i ; i ]; where k(H ) is a bounded function. Proof. It is not dicult to verify that the solutions x0 ; x1 of Eq. (3) and the solutions y0 ; y1 of Eq. (6) which we have used in Eqs. (4) and (7) remain in GH . Subtracting Eq. (4) from Eq. (7) we nd that Z i ˆ v; ) − Si (x; u; v; ) = {(I + Bi )A(t)(y0 (t) − x0 (t)) − A(t)(y1 (t) − x1 (t)) Si (x; u; i
ˆ ) − g(t; x0 (t); u(t); )) +[(I + Bi )(g(t; y0 (t); u(t); ˆ ) − g(t; x1 (t); u(t); ))]} dt −(g(t; y1 (t); u(t); Z i +Bi C(t)(u(t) ˆ − u(t)) dt i
+[Wi (y0 (i ); vi ; ) − Wi (x0 (i ); vi ; )] Z i − {(I + Bi )A(t)x0 (t) − A(t)x1 (t) + Bi C(t)u(t) i
+[(I + Bi )g(t; x0 (t); u(t); ) − g(t; x1 (t); u(t); )]} dt: (10) Since the right-hand side of Eq. (3) satis es a Lipschitz condition in u, there exists a constant l1 , see [5], such that |y0 (t) − x0 (t)| ≤ l1 ku − uk ˆ 1:
(11)
It follows that |g(t; y0 (t); u(t); ˆ ) − g(t; x0 (t); u(t); )| ≤ l(1 + l1 )ku − uk ˆ 1: On the other hand, since Eq. (8) is satis ed and i − i = i (x0 (i ); ) − i (y0 (i ); ) Z i [A(t)x0 (t) + C(t)u(t) + f(t) + g(t; x0 (t); u(t); )] dt ≤ l i ! ˆ 1 +l1 ku − uk
ˆ 1 ]; ≤ l[(i − i )(2m1 H + m3 + m2 ) + l1 ku − uk
(12)
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M.U. Akhmetov, A. Zafer / Nonlinear Analysis 34 (1998) 1055 – 1065
we have i − i ≤
ll1 ku − uk ˆ 1; 1 − lh1 (H; )
h1 (H; ) = 2m1 H + m3 + m2 :
(13)
Using Eq. (13) and the expression Z i [A(t)x1 (t) + C(t)u(t) + f(t) + g(t; x1 (t); u(t); )] dt; x1 (i ) = x1 (i ) + i
we nd that ˆ 1; |x1 (i ) − x1 (i )| ≤ h2 (H; )ku − uk
h2 (H; ) =
ll1 h1 (H; ) : 1 − lh1 (H; )
It follows that |y1 (t) − x1 (t)| ≤ l2 h2 (H; )ku − uk ˆ 1;
(14)
where l2 is a xed positive real number. Now, we shall estimate the integrals in Eq. (10). Denote them in order by I1 , I2 , and I3 . Using Eqs. (11), (12), and (14) we obtain |I1 | ≤ m3 {(1 + m1 )m1 l1 + m1 h2 (H; ) + [(1 + m1 )l(1 + l1 ) ˆ 1 = k 1 (H )ku − uk ˆ 1; +l(1 + h2 (H; ))]}ku − uk |I2 | ≤ m21 m3 ku − uk ˆ 1 = k 2 (H )ku − uk ˆ 1
(15) (16)
and |I3 | ≤
ll1 {(1 + m1 )m2 H + (m1 + m21 )H 1 − lh2 (H; ) ˆ 1: +[(1 + m1 )m3 + m3 ]} = k 3 (H )ku − uk
(17)
Moreover, |Wi (y0 (i ); vi ; ) − Wi (x0 (i ); vi ; )| ≤ l(l1 + h2 (H; ))ku − uk ˆ 1 = k 4 (H )ku − uk ˆ 1:
(18)
By summing relations (15)–(18) and noting that k i , i = 1; : : : ; 4, are bounded functions of H , we see that Eq. (9) holds. Thus, the lemma is proved. In what follows, we denote by X (t); X ( ) = I , a fundamental matrix of dx = A(t)x; t 6= i dt x|t=i = Bi x
M.U. Akhmetov, A. Zafer / Nonlinear Analysis 34 (1998) 1055 – 1065
and de ne (t) =
Z
t
Q(t)QT (t) dt +
X
1061
Pi PiT ;
¡i ¡t
where Q(t) = X −1 (t)C(t) and Pi = X −1 (i )Di . The problem when = 0 was considered in [2] and the following theorems were obtained. Theorem 1. The problem 0 is solvable if and only if the matrix ( ) is non-singular. Theorem 2. Let 0 be solvable. Then the pair {u; v} ∈ pm is a solving control for 0 if and only if it has the form ˆ u = QT (t)c + u(t); where
t ∈ [ ; ];
vi = PiT c + vˆi ;
"
c = −1 ( ) X −1 ( )b − X −1 ( )a −
Z
i = 1; : : : ; p;
X −1 (t)f(t) dt −
p X
# X −1 (i )Ji ;
i=1
and {u; ˆ v} ˆ ∈ pm is orthogonal to all columns of [QT ; PiT ]. 2. Control of systems with xed moments of impulse actions Consider the following system: dx = A(t)x(t) + C(t)u + f(t) + g(t; x; u; ); dt
t 6= i
x|t=i = Bi x + Di vi + Ji + Wi (x; vi ; )
(19)
with boundary condition (2). All quantities here are as in Eq. (1) except that the moments of impulse actions are now xed. We denote this problem by . Theorem 3. If the matrix ( ) is non-singular; then is solvable. The solving control of this problem is the limit of a uniformly convergent sequence obtained by the method of successive approximations. Proof. We will prove that is solvable with a control {u; v} of the following form: ˆ u = QT (t)c + u(t);
t ∈ [ ; ];
vi = PiT c + vˆi ;
i = 1; : : : ; p;
(20)
ˆ v} ˆ ∈ pm is orthogonal to all columns of where c ∈ Rn is a constant vector and {u; T T [Q ; Pi ].
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M.U. Akhmetov, A. Zafer / Nonlinear Analysis 34 (1998) 1055 – 1065
The problem is equivalent to solving Z t Q(s)(u(s) + C −1 (s)f(s) + C −1 (s)g(s; x(s); u(s); )) ds x(t) = X (t) a +
X
+
Pi (vi + Di−1 Ji + Di−1 Wi (x(i ); vi ; )) ;
x( ) = b:
(21)
¡i ¡t
Substituting Eq. (20) into Eq. (21) we obtain the vector c as " Z −1 −1 −1 X −1 (t)(f(t) + g(t; x(t); u(t); )) dt c = ( ) X ( )b − X ( )a −
p
−
X
#
X −1 (i ) (Ji + Wi (x(i ); vi ; )) :
(22)
i=1
If we let u0 (t) = QT (t) ( )−1 K + u(t); ˆ vi0 = PiT ( )−1 K + vˆi ; Z t X (Q(s)u0 (s) + X −1 (s)f(s)) ds + (Pi vi0 + X −1 (i )Ji ) ; x0 (t) = X (t) a +
¡i ¡t
0 = (x0 (t); u0 (t); vi0 ); = (x(t); u(t); vi ); Z t X −1 (s)g(s; x(s); u(s); ) ds; (t; ; ) =
(t; ; ) =
X
X −1 (i )Wi (x(i ); vi ; );
¡i ¡t
where K = X −1 ( )b − X −1 ( )a −
Z
X −1 (t)f(t) dt −
p X
X −1 (i )Ji ;
i=1
then, by substituting Eq. (22) into Eq. (21) we nd that satis es the following relation: = 0 + P( ; ); where P = (P1 ; P2 ; Pi ), P1 (t; ) = X (t)[(t; ; ) + (t; ; ) − (t) −1 ( )(( ; ; ) + ( ; ; ))]; P2 (t; ) = Q(t)T −1 ( )[( ; ; ) + ( ; ; )]; Pi ( ) = PiT −1 ( )[( ; ; ) + ( ; ; )]:
(23)
M.U. Akhmetov, A. Zafer / Nonlinear Analysis 34 (1998) 1055 – 1065
1063
We introduce the norm k · k, k k = max |x(t)| + max |u(t)| + max |vi |; t
t
i
in the space of all elements of the form = (x(t); u(t); vi ). Suppose that the number h which is xed earlier satis es h¿k 0 k; and that 4 is a positive real number such that 4 ≤ 1 and 4 max kP( ; )k¡h − k 0 k: k k≤h 0¡≤1
Let be the subspace consisting of elements = (x(t); u(t); vi ) such that k k ≤ h, where x(t) is piecewise absolutely continuous, continuous on the left, and has discontinuities of the rst kind at moments i ; i = 1; : : : ; p; and {u; v} ∈ pm . It follows that if ≤ 4 then the operator 0 + P( ; ) maps to itself. We shall show that if is suciently small then P is a contraction mapping. Let m4 = max max |Q(t)T −1 ( )X −1 (s)|; max |PiT −1 ( )X −1 (s)|; t; s
max |X (t)X t; s
i; s
−1
−1
(s)|; max |X (t) (t) t; s
( )X
−1
(s)| :
Then, we have kP( 1 ; ) − P( 2 ; )k ≤ 4m4 l( − + p)k 1 − 2 k uniformly with respect to t ∈ [ ; ], ∈ [0; 4 ], and i = 1; : : : ; p for any 1 ; 2 ∈. Now if we let 0 ¡4 be such that 40 m4 l( − + p)¡1; then for ¡0 , P becomes a contraction mapping. Thus, we can obtain a sequence { i }, i = 0; 1; 2; : : : ; i+1 = 0 +P( i ; ) converging to some element 0 ∈ , 0 = (x0 ; u0 ; v0 ). Substituting 0 in Eq. (19), di erentiating x0 (t), and checking the boundary condition (2) and the condition of discontinuities, one can verify that 0 solves . Thus, the proof is complete. 3. Control of systems with variable moments of impulse actions Now, we return to considering problem . In view of the property of systems (1) and (5), the problem is reduced to a problem of the kind for system (5). Let us denote it by . The following lemma is concerned with the solvability of .
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M.U. Akhmetov, A. Zafer / Nonlinear Analysis 34 (1998) 1055 – 1065
Lemma 3. If ( ) is non-singular then is solvable. Proof. Let ¡3 , where 3 is as in Eq. (8), and let (x; u; v) ∈ h . Then using Eq. (4) we obtain |Si (x; u; vi ; )| ≤m5 ; where m5 = m3 [(2 + m1 )(2m1 h + m2 + 3 m3 ) + 3 m3 ]. Since Si is di erentiable in x and vi , there is a positive real number 5 such that for ¡5 it satis es a Lipschitz condition in x and v. The corresponding Lipschitz constants have the form k2 (H ) and k3 (H ), where k2 and k3 are bounded functions. Setting k(H ) = max{k1 ; k2 ; k3 } and 6 = min{3 ; 5 }, we see, in view of Lemma 2, that if ¡6 then |Si (x1 ; u1 ; v1 ; ) − Si (x2 ; u2 ; v2 ; )| ≤ k(H ){|x1 − x2 | + ku1 − u2 k1 + |v1 − v2 |} uniformly in h . Now, we let 0 ∈ be as in Theorem 3 and consider the operator = 0 +T( ; ) in , where T = (T1 ; T2 ; Ti ), T1 ( ; ) = X (t)[(t; ; ) + ˜ (t; ; ) − (t) −1 ( )(( ; ; ) + ˜ ( ; ; ))]; T2 ( ; ) = Q(t)T −1 ( )[( ; ; ) + ˜ ( ; ; )]; Ti ( ; ) = PiT −1 ( )[( ; ; ) + ˜ ( ; ; )] with (t; ; ) as de ned in Eq. (23) and X ˜ (t; ; ) = 1 X −1 (i )Si (x(i ); u; vi ; ): ¡i ¡t
Choosing 0 as 0 = min 6 ;
4(h − k 0 k) 4 ; ( − )m3 m4 + pm4 m5 lm4 ( − ) + k(H )m4p
;
we can see similarly as in the proof of Theorem 3 that if ¡0 , the operator 0 + T( ; ) maps to itself and is contractive. Thus, the lemma is proved. Theorem 4. If () is non-singular then is solvable. Proof. Lemma 3 implies that there is a control {u; ˆ v} ˆ such that system (5) has a solution y(t) which satis es y( ) = a and y( ) = b. Since Eqs. (1) and (5) have the property , system (1) with the control {u; ˆ v} ˆ admits a solution x(t) such that x(t) = y(t) for all t except for t ∈ [i ; i ], i = 1; : : : ; p. Since and are not in [i ; i ], i = 1; : : : ; p, the solution x(t) also satis es the boundary condition (2). This completes the proof.
M.U. Akhmetov, A. Zafer / Nonlinear Analysis 34 (1998) 1055 – 1065
1065
References [1] M.U. Akhmetov, N.A. Perestyuk, On comparison method for di erential systems with pulse e ect, Di erential Equations 9 (26) (1990) 1079 –1086. [2] M.U. Akhmetov, N.A. Perestyuk, M.A. Tleubergenova, The control of linear impulsive systems, Ukrainian. Math. Zh. 3 (47) (1995) 307–314. [3] S.A. Aysagaliev, K.O. Onaybaer, T.G. Mazakov, The controllability of nonlinear systems, Izv. Akad. Nauk. Kazakh.-SSR.-Ser. Fiz.-Mat (1) (1985) 83 –87. [4] S. Barnett, Introduction to Mathematical Control Theory, Clarendon Press, Oxford, 1975. [5] P. Hartman, Ordinary Di erential Equations, Wiley, New York, 1964. [6] A. Halanay, D. Wexler, Qualitative Theory of Impulse Systems, Editura Academiei Republici Socialiste Romania, Bucuresti, 1968. [7] N.N. Krasovsky, Theory of Motion Control, Linear Systems, Nauka, Moscow, 1973. [8] V. Lakshmikantham, D.D. Bainov, P.S. Simeonov, Theory of Impulsive Di erential Equations, World Scienti c, Singapore, 1989. [9] E.B. Lee, L. Markus, Foundations of Optimal Control Theory, Wiley, New York, 1967. [10] A.M. Samoilenko, N.A. Perestyuk, Impulsive Di erential Equations, World Scienti c, Singapore, 1995.