Nonlinear Oscillations, Vol. 10, No. 4, 2007 CLOSEDNESS AND NORMAL SOLVABILITY OF AN OPERATOR GENERATED BY A DEGENERATE LINEAR DIFFERENTIAL EQUATION WITH VARIABLE COEFFICIENTS S. M. Zhuk1 UDC 517.9 n m m For a linear operator D : WF generated by the differential equation 2 ⊂ L2 → L2 × R d F x(t) − C(t)x = f (t), dt F x(t0 ) = f0 , 0 m m we prove that its graph is closed and determine the adjoint operator D∗ : WF → Ln 2 ⊂ L2 × R 2 . For F F0 elements of the linear manifolds W2 and W2 , we propose an analog of the formula of integration by parts. We establish a criterion for the existence of a pseudosolution of the operator equation Dx(·) = (f (·), f0 ) and formulate sufficient conditions for the normal solvability of the operator D in terms of relations for blocks of the matrix C(t). The results obtained are illustrated by examples. 1. Introduction In the process of solution of various problems in applied fields such as mathematical economics, robotics, biotechnology, digital image processing, control theory, circuit theory, radiophysics, and chemical and biological kinetics, researchers encounter systems whose state is described by differential equations with degeneracy: F (t)ẋ(t) + C(t)x(t) + B(t)f (t) = 0. (1) In literature, systems of the form (1) are called degenerate [1], algebraic–differential [2], singular [3–5], descriptor2 [6–9], and implicit [10] or unresolved with respect to the derivative [11]. For autonomous systems (1) in the case where the pencil of matrices λF + C is regular,3 the notion of central canonical form proposed in [5] proved to be fruitful. Namely, for a smooth vector function f (·), a stationary regular system can be reduced to the following form by a nondegenerate linear transformation [7, p. 14]: ẋ1 (t) = Ax1 (t) + Kf (t), x2 (t) = −Df (t) − m−1 X i=1 i (i) N Dv (t)), x1 = Q−1 x(t). x2 Sufficient conditions for the reducibility of (1) to a central canonical form were proposed in [1]. Note that the theory of singular matrix pencils was used as early as in [12, p. 348] for the description of the set of solutions of stationary singular linear differential equations. 1 Shevchenko Kyiv National University, Kyiv. In the present paper, we use the term “descriptor equation” or “descriptor system” to denote systems of the type (2). The emerging differences are described in what follows (see Remark 1). 3 A pencil of matrices λF + C is called regular if det(λ0 F + C) 6= 0 for a certain real λ0 . 2 Translated from Neliniini Kolyvannya, Vol. 10, No. 4, pp. 464–480, October–December, 2007. Original article submitted September 18, 2007. 1536–0059/07/01004–0469 c 2007 Springer Science+Business Media, Inc. 469 S. M. Z HUK 470 In the case of variable coefficients, the notion of central canonical form gives information on the structure of a general solution of the system only in special cases, e.g., [2, p. 8] for rankF (t) = deg det(λF (t) + C(t)) = const, t ∈ [t0 , T ]. In [2, 3], the problem of the existence of solutions of initial- and boundary-value problems for Eqs. (1) was studied on the basis of the notion of left (right) regularizing operator [2]: Λ∗,r [F (t)ẋ(t) + C(t)x(t) + f (t)] = ẋ(t) + Λ∗,r [C(t)]x(t) + Λ∗,r [f (t)], where Λ∗,r = r X Lj (t) j=0 d dt i . The existence of a left regularizing operator for autonomous systems [2, p. 11] is equivalent to the existence of a central canonical form. In the case of variable coefficients, conditions for the existence of a left regularizing operator [2] are connected with properties of the “extended” system (1), i.e., the system that consists of Eq. (1) and its total time derivatives up to a certain order. Attention of some investigators is focused on the study of qualitative problems in the theory of descriptor systems with operator coefficients in Banach spaces. The behavior of a descriptor system with delay in a Banach space was investigated in [13] on the basis of analysis of poles of the resolvent of the operator pencil R(t, λ) = (λF (t) + C(t))−1 . For detailed information on problems in the theory of descriptor systems such as the existence and uniqueness of solutions, construction of numerical methods, control and observability, see [6, 8]. 2. Statement of the Problem In most papers presented above, system (1) was studied on the basis of notions of central canonical form and regularizing operators, i.e., the construction of solutions was based on the reduction of Eq. (1) to a certain canonical form, which enables one to use methods developed for normal ordinary differential equations. In this case, the structure of the corresponding descriptor equation is restricted. In the present paper, we use an alternative approach based on the investigation of the properties of an operator generated by the linear descriptor equation d F x(t) − C(t)x(t) = f (t), dt (2) F x(t0 ) = f0 without assuming the possibility of its reduction to a certain canonical form. Operator methods enable one to use general properties of a broad class of descriptor systems, including so-called “noncausal” systems (see [6, 10]). In Eq. (2), let F be an m × n matrix, let t 7→ C(t) be a continuous m × n matrix-valued function, and let f (·) ∈ L2 ([t0 , T ], Rm ) := Lm 2 , −∞ < t0 < T < +∞, f0 ∈ Rm . C LOSEDNESS AND N ORMAL S OLVABILITY OF AN O PERATOR 471 It is known that, in the case F = E, a vector function x(·) ∈ Ln2 that satisfies the Volterra integral equation of the second kind Zt x(t) = f0 + (C(s)x(s) + f (s))ds t0 is a solution of Eq. (2). An equation of this type has a unique solution x(·) in the class of absolutely continuous functions, and, furthermore, x(·) satisfies Eq. (2) almost everywhere. Let us clarify what we mean by a solution of (2) in the case of an arbitrary constant rectangular matrix F = {Fij }m,n 1,1 . We introduce the set WF2 := {x(·) ∈ Ln2 : F x(·) ∈ Wm 2 }, m where Wm 2 denotes the collection of all absolutely continuous vector functions from L2 whose derivatives lie m in L2 , and F x is an m-vector function whose ith component is a linear combination of components of x(·) with coefficients {Fij }nj=1 . The linear set WF2 is everywhere dense in Ln2 because Wn2 ⊂ WF2 . By analogy, we introduce the set 0 0 n WF2 := {z(·) ∈ Lm 2 : F z(·) ∈ W2 }. We define a linear operator D by the relations Dx(·) = d F x(·) − Cx(·), F x(t0 ) , dt x(·) ∈ D(D) := WF2 , d m F x(·) ∈ Lm 2 is the derivative of the vector function F x(·), Cx(·) ∈ L2 is a vector function t 7→ dt d C(t)x(t), and F x(t0 ) is the value4 of t 7→ F x(t) for t = t0 . Note that, for x(·) ∈ WF2 , the equality F x(t) = dt F ẋ(t) can be absent. In the case of an arbitrary constant rectangular matrix F, a solution of (2) is understood as an element of the set WF2 that satisfies the operator equation where Dx(·) = (f (·), f0 ). The operator D is defined in a similar way as in S. Krein’s work [14] for boundary-value problems for linear differential equations of order n. It follows from the structure of the introduced operator equation that the vector function x(·) from the space Ln2 belongs to the set of solutions of (2) for the fixed initial condition f0 and right-hand side f (·) if the vector function F x(·) is absolutely continuous and has the derivative of the class Lm 2 satisfying the first equality in (2) almost everywhere and, furthermore, the second equality in (2) is true. Descriptor equations of the form (2) were considered in [9, 11]. The aim of the present paper is to investigate properties of the operator D, namely, its closedness and normal solvability. In terms of descriptor systems, we investigate the problem of the generalized solvability of Eq. (2), conditions for the continuous (in the metric of the corresponding Hilbert space) dependence of a pseudosolution on the right-hand side and initial condition, and the approximation of solutions by elements of a regularizing sequence. The symbol F x(t) makes sense for any t ∈ [t0 , T ] because, by virtue of the inclusion x(·) ∈ WF 2 , the vector function t 7→ F x(t) is absolutely continuous, which, in the general case, cannot be stated for x(·). 4 S. M. Z HUK 472 3. Closedness of the Operator D. Form of the Adjoint Operator D ∗ Generally speaking, the introduced operator D is not a Fredholm one (Example 3), and, moreover, it is not even normally solvable (Example 2) unlike the operator of ordinary differentiation in Ln2 . However, in the general case, it is densely defined and closed (Theorem 1). 0 Theorem 1. For x(·) ∈ WF2 and z(·) ∈ WF2 , the following analog of the formula of integration by parts is true: ZT d 0 d F x(t), z(t) + F z(t), x(t) dt = (F x(T ), F 0+ F 0 z(T )) − (F x(t0 ), F 0+ F 0 z(t0 )). dt dt (3) t0 The operator D is closed, and its adjoint D∗ is determined by the relations D∗ (z(·), z0 ) = L(z(·), z0 ) := − d 0 F z(·) − C 0 z(·), dt 0 D(D∗ ) = D(L) := {(z(·), F 0+ F 0 z(t0 ) + d) : z(·) ∈ WF2 , F 0 z(T ) = 0, F 0 d = 0}. 0 Proof. We choose x(·) ∈ WF2 and z(·) ∈ WF2 and use the formula of integration by parts for the absolutely continuous vector functions F + F x(·) and F 0 z(·). We obtain ZT d + d F F x(t), F 0 z(t) + F + F x(t), F 0 z(t) dt = F + F x(T ), F 0 z(T ) − F + F x(t0 ), F 0 z(t0 ) . dt dt t0 By virtue of the Moore–Penrose theorem, we get F F + F = F. Therefore, almost everywhere, we have F d + d d F F x(t) = F F + F x(t) = F x(t), dt dt dt whence ZT t0 ZT ZT d + d d 0 + F F x(t), F z(t) dt = F F F x(t), z(t) dt = F x(t), z(t) dt dt dt dt t0 t0 for every absolutely continuous z ∈ Lm and, hence, for any z ∈ Lm 2 2 . We prove the theorem in the case5 C(t) ≡ 0. Let us show the closedness of D. To this end, we note that the d operator x(·) 7→ F x(·), x(·) ∈ WF2 , is closed. dt 5 A generalization is carried out by standard reasoning on the closedness of the sum of closed and bounded operators and on the operator adjoint to this sum. C LOSEDNESS AND N ORMAL S OLVABILITY OF AN O PERATOR 473 d F xn (·) → z(·),6 then F xn (·) is absolutely continuous and F xn (·) → F x(·). dt Taking into account that the operator of differentiation is closed in Lm 2 , we establish that F x(·) is absolutely d continuous and F x(·) = z(·). dt d F x(·) = z(·), and, according to the Let Dxn (·) → (z(·), z0 ) and xn (·) → x(·). Then x(·) ∈ WF2 , dt results proved above, it remains to show that z0 = F x(t0 ). Indeed, by assumption, we have F xn (t0 ) → z0 . On the other hand7 Indeed, if xn (·) → x and t Z d d → 0, kF x(t0 ) − F xn (t0 )k ≤ JkF xn (·) − F x(·)k2 + sup F x(s) − F x (s) ds n dt dt t∈[t0 ,c] t0 1 where J := (c − t0 )− 2 . We have shown the closedness of the densely defined operator D, which proves the existence of D∗ [15, p. 40]. Let us show that the other statements are also true. Let (z(·), z0 ) ∈ D(L) and x(·) ∈ D(D). Then ZT hDx(·), (z(·), z0 )i1 = d F x(t), z(t) dt + F x(t0 ), F 0+ F 0 z(t0 ) dt t0 0+ ZT 0 = (F x(T ), F F z(T )) − d 0 F z(t), x(t) dt = hL(z(·), z0 ), x(·)i2 dt t0 and, according to (3), L ⊂ D∗ . It remains to prove that D(D∗ ) ⊂ D(L). Indeed, let v(·) = D∗ (z(·), z0 ) for a certain (z(·), z0 ) ∈ D(D∗ ). Then v(·) ⊥ N (D). Since + F (E − F F ) = 0 ⇒ d + + F (E − F F )x(·), F (E − F F )x(t0 ) = (0, 0) ∀x(·) ∈ Wn2 , dt we conclude that v(·) = F + F v(·) almost everywhere. In this case, ZT t0 ZT ZT ZT d d F x(t), z(t) dt = F 0+ v(t), F x(t) dt = F 0+ v(s)ds, F x(t) dt dt dt t0 t0 t for all x(·) ∈ M := {x(·) ∈ WF2 : F x(t0 ) = 0}, and, hence, ZT z(·) − F 0+ v(s)ds ⊥ R(L0 ), t 6 7 1/2 Convergence is understood in the sense of the norm k · k2 := h·, ·i2 The quantity k · k is the norm in the space Rn . of the space Ln 2. S. M. Z HUK 474 where L0 denotes the operator x(·) 7→ d F x(·), x(·) ∈ M. If dt Zt g(t) = F F + ϕ(s)ds, ϕ(·) ∈ Lm 2 , t0 then L0 g(·) = F F + ϕ(·) and, hence, F F + ϕ(·) ∈ R(L0 ) for any ϕ(·) ∈ Lm 2 , which yields ZT ZT z(t) − t0 F 0+ v(s)ds, F F + ϕ(t) dt = 0 ∀ϕ(·) ∈ Lm 2 . t In this case, we have F 0 z(t) − ZT F 0+ v(s)ds = F 0 (E − F 0+ F 0 ) z(t) − t ZT F 0+ v(s)ds = 0, t 0 i.e., z(·) ∈ WF2 and F 0 z(T ) = 0. Thus, for any x(·) ∈ WF2 , we get ZT d 0 d F x(t), z(t) + F z(t), x(t) dt = −(F x(t0 ), F 0+ F 0 z(t0 )) dt dt t0 by virtue of (3). On the other hand, ZT ZT d F x(t), z(t) dt + (F x(t0 ), z0 ) = (D∗ (z(·), z0 )(t), x(t))dt dt t0 t0 and, hence, ZT d ∗ (F x(t0 ), z0 − F F z(t0 )) = D (z(·), z0 )(t) + F x(t), x(t) dt dt 0+ 0 t0 for all x(·) ∈ WF2 and, in particular, for x(·) ∈ M (see the definition of M given above). Taking into account that cl M = Ln2 , we obtain (F x(t0 ), z0 − F 0+ F 0 z(t0 )) = 0 for all x(·) ∈ WF2 . It is now clear that z0 = F 0+ F 0 z(t0 ) + d and F 0 d = 0. The theorem is proved. C LOSEDNESS AND N ORMAL S OLVABILITY OF AN O PERATOR 475 d Remark 1. Generally speaking (see Example 1), the operator x(·) 7→ F x(·), x(·) ∈ Wn2 , is not closed. dt In particular, this means that ∗ d d F0 ⊂ F . dt dt d On the other hand, the space Wn2 is not always a Hilbert space with respect to the norm of the graph of F. dt d Indeed, if Wn2 is a Hilbert space, then the restriction of F to Wn2 is a closed operator [15, p. 26]. However, dt d d F x(·) = F x(·) dt dt i.e., F ∀x(·) ∈ Wn2 , d d is the restriction of F to Wn2 . dt dt Example 1. Let t0 = 0, T = 1, n = 2, and 1 0 F = . 0 0 d defined on Wn2 . The Cantor function t 7→ k(t) is continuous and almost everywhere dt differentiable on [0, 1], but not absolutely continuous. Therefore, (0, k(·)) 6∈ Wn2 . On the other hand, according to the Bernstein theorem, the function k(·) can be approximated uniformly in t by the polynomials Consider the operator F n X i n Bn (t) := k ti (1 − t)n−i . i n 0 For each n, the polynomial Bn (·) is an absolutely continuous function. Therefore, xn (·) := 0 ∈ Wn2 . Bn (·) d On the other hand, F xn (·) = (0, 0) for any n, and xn converges to x(·) := (0, k(·)) in Ln2 , which implies dt that xn (·) → x(·) and F d xn (·) → (0, 0) dt d d is closed, then x(·) ∈ Wn2 by virtue of the closedness of the graph of F dt dt d [15, p. 17]. However, this inclusion is not true by virtue of the choice of x(·). Consequently, the operator F is dt not closed. d On the other hand, x(·) ∈ WF2 . Therefore, for the operator F defined on WF2 , we get dt simultaneously. If the operator F xn (·) → x(·), d F xn (·) → (0, 0) ⇒ x(·) ∈ WF2 , dt d F x(·) = (0, 0). dt S. M. Z HUK 476 4. Conditions for the Normal Solvability of the Operator D The property of normal solvability of the operator of a system is very important for various applications of linear differential equations. This property enables one to speak of the continuous dependence of a solution with the least norm on the right-hand side. The fact that the operator D is closed and densely defined enables one to use methods of the theory of ill-posed problems [16, 17] for the investigation of the problem of the normal solvability of D (Theorems 2 and 3). The theorem presented below gives a criterion for the existence of a pseudosolution8 of Eq. (2) for given (f (·), f0 ). Theorem 2. The boundary-value problem d F x(t) = C(t)x(t) + z(t) + f (t), dt d 0 F z(t) = −C 0 (t)z(t) + ε2 x(t), dt F 0 z(T ) = 0, F x(t0 ) − F 0+ F 0 z(t0 ) − d = f0 , (4) F 0 d = 0, m has a unique solution (x(·, ε), z(·, ε), d(ε)) for every ε > 0. For arbitrary (f (·), f0 ) ∈ Lm 2 × R , the descriptor equation d F x(t) = C(t)x(t) + f (t), dt F x(t0 ) = f0 , (5) has a pseudosolution x b(·) if and only if kx(·, ε)k2 ≤ C as ε → 0. m and show that (4) has a unique solution. We choose ε > 0 and Proof. We fix g(·) := (f (·), f0 ) ∈ Lm 2 ×R 1 consider the operator D. The problem of the projection ε 2 1 Dp(·) − g(·) + kp(·)k2 → 2 ε 1 min p(·)∈D(D) (6) 1 of the vector (0, g(·)) to the graph of the operator D has a unique solution p̂(·) ∈ D(D) by virtue of the ε 1 closedness of D because the product of the bounded operator of multiplication by a scalar and a closed operator ε is a closed operator. Since9 2 1 Dp(·) − g(·) + kp(·)k22 = kDx(·) − g(·)k21 + ε2 kx(·)k22 ε 1 Here, a pseudosolution is understood as a vector x̂(·) ∈ D(D) with the least norm for which hDx̂(·) − (f (·), f0 ), Dx(·)i = 0 ∀x(·) ∈ D(D). 9 m The symbol k · k1 denotes the norm in the Hilbert space Lm 2 ×R . 8 C LOSEDNESS AND N ORMAL S OLVABILITY OF AN O PERATOR 477 for p(·) = εx(·), by virtue of the arguments presented above the optimization problem kDx(·) − g(·)k21 + ε2 kx(·)k22 → min x(·)∈D(D) has a unique solution x̂ε (·) ∈ D(D) that satisfies the system of operator equations Dx(·) − z(·) = g(·), −D∗ z(·) − ε2 x(·) = 0. It is easy to verify that this system is equivalent to (4). The function L (ε) := min x(·)∈D(D) kDx(·) − g(·)k21 + ε2 kx(·)k22 = kDx̂ε (·) − g(·)k21 + ε2 kx̂ε (·)k22 satisfies the relation [16, p. 119] (Lemma 1.25) L (0+) = L (0) = min x(·)∈D(D) kDx(·) − g(·)k21 . Thus, lim kDx̂ε (·) − g(·)k21 = ε→0 min x(·)∈D(D) kDx(·) − g(·)k21 . (7) It is known [17, p. 124] (Theorem 18.5) that the convergence of the sequence {x̂ε (·)} as ε → 0 is equivalent to the existence of x̂(·) ∈ D(D) such that kDx̂(·) − g(·)k21 = min kDx(·) − g(·)k21 . x(·) Let f (·) ∈ cl R(D). Then Dx̂ε (·) → f (·), and the set {Dx̂ε (·)} is bounded. If {x̂ε (·)} is also bounded, then, by virtue of the weak closedness of D, we have f (·) ∈ R(D). Otherwise, for f (·) ∈ / R(D), the norms of the functions {x̂ε (·)} increase unboundedly as ε → 0 by virtue of (7). In the context of the theorem proved, the following lemma may also be useful: Lemma 1. For a rectangular m × n matrix F, there exist square matrices L and R such that F = LΛR, F + = R0 Λ+ L0 , LL0 = Em , RR0 = En , 1 D2 0r,n−r , Λ= 0m−r,r 0m−r,n−r (8) where Em is the identity matrix of order m, 0k,s is the zero k × s matrix, and D is the diagonal matrix of order r formed by the positive eigenvalues λ1 , . . . , λr of the matrix F F 0 . S. M. Z HUK 478 Proof. Using the theorem on the singular-value decomposition of a rectangular matrix F (see [18, p. 52]), 1 we write F = P D 2 Q, where P and Q are, respectively, m × r and r × n rectangular matrices and P 0 P = QQ0 = Er . We decompose the matrix Pr,r P = ! Pm−r,r into blocks. The columns of the matrix P are orthonormal because P 0 P = Er . Using the Gram–Schmidt orthogonalization, we can complement them by m − r vectors wk to the orthonormal basis of the corresponding Euclidean space. We represent the matrix T whose columns are the m − r vectors wk in the form Tr,m−r T = Tm−r,m−r ! . Then T 0 P = 0 and T 0 T = Em−r by construction. We set L := Pr,r Tr,m−r ! . Pm−r,r Tm−r,m−r Then L0 L = Em ⇒ LL0 = Em because P 0 P = Er , T 0 P = 0, and T 0 T = Em−r . By analogy, we define R := Qr,r Qr,n−r ! , Wn−r,r Wn−r,n−r where Q = Qr,r Qr,n−r and W = Wn−r,r Wn−r,n−r is a matrix whose rows are n − r vectors vk that complement the orthonormal row vectors of the matrix Q to the orthonormal basis of the corresponding Euclidean 1 space. Thus, RR0 = En . One can directly verify that RΛL = P D 2 Q. Let us show that F + = R0 Λ+ L0 . We determine (LΛR)+ . It is known [18, p. 69] that (AB)+ = B + A+ if and only if R(BB 0 A0 ) ⊆ R(A0 ) and, simultaneously, R(A0 AB) ⊆ R(B) for rectangular matrices A and B. According to the results proved above, we have L0 L = Em . Therefore, R(ΛRR0 ΛL0 ) ⊂ R(L0 ) and R(L0 LΛR) = R(ΛR), whence (LΛR)+ = (ΛR)+ L0 because L+ = L−1 = L0 . Since RR0 = En , we conclude that (ΛR)+ = R0 Λ+ because R(RR0 Λ0 ) = R(Λ0 ) and R(Λ0 ΛR) ⊂ R(R). The lemma is proved. In the theorem below, a class of equations of special structure that generate a normally solvable operator is separated from the singular systems (2). Theorem 3. Suppose that 0 0 L C(t)R = C1 C2 C3 C4 ! , C LOSEDNESS AND N ORMAL S OLVABILITY OF AN O PERATOR 479 where L and R are the same as in Lemma 1 and Ci are stationary rectangular matrices of consistent dimensions. If 10 sup kQ(ε)C20 kmod < +∞, 1>ε>−1 Q(ε) := (ε2 E + C40 C4 )−1 , then the operator D has a closed range of values. Proof. Let us show that, under the conditions of the theorem, the norms of solutions of the boundary-value problem (4) remain bounded as ε → 0. To this end, we rewrite (4) in the form d dt F 0 0 F0 x(t) C(t) E x(t) f (t) = + . z(t) ε2 −C 0 (t) z(t) 0 Using the equality (see Lemma 1) 0 Λ 0 L 0 F = 0 0 Λ 0 R 0 0 F0 0 R 0 , 0 L we obtain the equivalent equation d dt 0 p(t) L C(t)R0 E p(t) g(t) Λ 0 = + , q(t) ε2 −RC 0 (t)L q(t) 0 0 Λ0 where p(t) := Rx(t), q(t) := L0 z(t), and g(t) := L0 f (t). The conditions F 0 z(T ) = 0, F x(t0 ) = F 0+ F 0 z(t0 ) + d + f0 , and F 0 d = 0 take the form Λ0 q(T ) = 0, + Λp(t0 ) = Λ0 Λ0 q(t0 ) + L0 d + L0 f0 , Λ0 L0 d = 0. Taking into account the form of Λ, we obtain the system of algebraic–differential equations 1 1 D 2 ṗ1 (t) = C1 p1 (t) + C2 (t)p2 (t) + g1 (t) + D− 2 v1 (t), 0 = C3 p1 (t) + C4 p2 (t) + g2 (t) + q2 (t), 1 v̇1 (t) = ε2 p1 (t) − C10 D− 2 v1 (t) − C30 q2 (t), 1 0 = ε2 p2 (t) − C20 D− 2 v1 (t) − C40 q2 (t), 1 p1 (t0 ) − D−1 v1 (t0 ) = D− 2 g10 , 10 For a matrix F, we set kF kmod := P i,j |Fij |. d2 = g20 , v1 (T ) = 0, (9) S. M. Z HUK 480 where 1 2 v1 (t) := D q1 (t), p(t) := p1 (t) , p2 (t) 0 g g0 := L f0 = 10 , g2 0 q1 (t) q(t) := , q2 (t) 0 Ld= . d2 0 We write the algebraic equations separately: C4 E ! ε2 E −C40 ! p2 (t) q2 (t) = −C3 p1 (t) − g2 (t) 1 C20 D− 2 v1 (t) ! . (10) Multiplying (10) from the left by Q(ε)C40 Q(ε) ! , E − C4 Q(ε)C40 −C4 Q(ε) we get 1 p2 (t) = Q(ε)(−C40 C3 p1 (t) + C20 D− 2 v1 (t) − C40 g2 (t)), (11) q2 (t) = (E − C4 Q(ε)C40 )(−C3 p1 (t) − g2 (t)) − C4 Q(ε)C20 D − 12 v1 (t). Substituting the obtained relations into (9), we get the following two-point boundary-value problem for a nonnegative-definite Hamiltonian system with parameter: ṗ1 (t) = A(ε)p1 (t) + R(ε)v1 (t) + g(t, ε), p1 (t0 ) − D−1 v1 (t0 ) = g0 , (12) v̇1 (t) = −A0 (ε)v1 (t) + S(ε)p1 (t) + `(t, ε), v1 (T ) = 0, where 1 g0 := D− 2 g10 , `(t, ε) = C30 (E − C4 Q(ε)C40 )g2 (t), 1 A(ε) = D− 2 (C1 − C2 Q(ε)C40 C3 ), S(ε) = ε2 E + C30 (E − C4 Q(ε)C40 )C3 , 1 1 R(ε) = D−1 + D− 2 C2 Q(ε)C20 D− 2 , 1 g(t, ε) = D− 2 (g1 (t) − C2 Q(ε)C40 g2 (t)). Using the theorem on “lower” and “upper” solutions [19], we can show that the Riccati equation K̇(t) = A(ε)K(t) + K(t)A0 (ε) − K(t)S(ε)K(t) + R(ε), K(t0 ) = D−1 , (13) C LOSEDNESS AND N ORMAL S OLVABILITY OF AN O PERATOR 481 has a unique solution t 7→ K(t, ε) defined on [t0 , T ] for every ε > 0, and, furthermore, K(t, ε) is a nonnegativedefinite matrix in its domain of definition. Let q(·, ε) and ϕ(·, ε) be determined as solutions of the Cauchy problem ϕ̇(t) = (A(ε) − K(t, ε)S(ε))ϕ(t) + g(t, ε) − K(t, ε)`(t, ε), q̇(t) = (−A0 (ε) + S(ε)K(t, ε))q(t) + S(ε)ϕ(t) + `(t, ε), ϕ(t0 ) = g0 , (14) q(T ) = 0. (15) Performing the substitution, one can verify that the functions p1 (t, ε) = K(t, ε)q(t, ε) + ϕ(t, ε), v1 (t, ε) = q(t, ε) (16) satisfy (12). Let us show that, under the conditions of the theorem, the norm of ϕ(·, ε) remains bounded as ε → 0. Indeed, ϕ(t, ε) = Φ(t, t0 , ε)f10 Zt Φ(t, s, ε)(g(s, ε) − K(s, ε)`(s, ε))ds, + t0 where Φ(t, s, ε) is the normalized fundamental matrix for (14) for fixed ε > 0. It remains to show the boundedness of the functions that are elements of the matrices Φ(t, s, ε) and K(t, ε) and vectors g(s, ε) and `(s, ε) as ε → 0. 1 We set g(t) := D− 2 (g1 (t) − C2 C4+ g2 (t)) and `(t) := C30 (E − C4 C4+ )g2 (t). Then, as ε → 0, we get ZT (g(t, ε) − g(t), g(t, ε) − g(t)) dt t0 ZT = (D−1 (C2 C4+ − C2 Q(ε)C40 )g2 (t), (C2 C4+ − C2 Q(ε)C40 )g2 (t))dt → 0 t0 because Q(ε)C40 → C4+ as ε → 0. Hence, there exist g, ` > 0 such that kg(·, ε)k2 ≤ g, k`(·, ε)k2 ≤ `, ε → 0. We introduce a function Kl (·, ε) as a solution of the Bernoulli equation K̇l (t) = A(ε)Kl (t) + Kl (t)A0 (ε) + R(ε), Kl (t0 ) = D−1 . Differentiating the function t 7→ ((Kl (t, ε) − K(t, ε))p, p), we obtain (Kl (t, ε)p, p) ≥ (K(t, ε)p, p), t0 ≤ t ≤ T, ε > 0, S. M. Z HUK 482 which implies that sp (Kl (t, ε) ≥ sp (K(t, ε)). Let kF ksp = sp(F F 0 ). Then, for a certain U > 0, we have 1 kK(t, ε)kmod ≤ kK(t, ε)ksp ≤ sp(K(t, ε)) ≤ kKl (t, ε)kmod U because K(t, ε) is a nonnegative-definite symmetric matrix for t ≥ t0 and ε > 0. Taking into account that 1 kA(ε) − Akmod → 0 as ε → 0 for A := D 2 (C1 − C2 C4+ C3 ) and using the representation Kl (t, ε) = exp((A(ε) + A0 (ε))(t − t0 )) + Zt exp(A(ε)(t − s))R(ε) exp(A0 (ε)(t − s))ds, t0 one can easily establish that the following relations hold for sufficiently small ε > 0: kKl (t, ε)kmod ≤ k exp((A + A0 )(t − t0 ))kmod + 1, Zt M (k exp(A(t − s))kmod + 1)(1 + k exp(A0 (t − s))kmod )ds := K 1 (t), t0 where kR(ε)kmod ≤ M. Thus, kK(t, ε)kmod ≤ U K 1 (t). Setting P (t, ε) := A(ε) − K(t, ε)S(ε), we get kP (t, ε)kmod ≤ 1 + kAkmod + U K 1 (t)(kC30 (E − C4 C4+ )C3 kmod + 1). Using this result, one can easily establish (see, e.g., [12, p. 432]) the boundedness of the elements Φ(t, s, ε) as ε → 0. By analogy, we prove the boundedness of q(·, ε). Using equalities (10) and (15), we establish the boundedness of p1 (·, ε) and p2 (·, ε) as ε → 0. Taking into account that x(t, ε) = R0 p(t, ε) and using Theorem 2, we obtain the required statement. We illustrate Theorem 2 by an example of a descriptor system of special form that generates an injective operator D with nonclosed range of values. Example 2. We set 1 0 F = , 0 0 1 −1 C(t) ≡ . 1 0 C LOSEDNESS AND N ORMAL S OLVABILITY OF AN O PERATOR 483 It is easy to verify that, in this case, the operator D is injective. Hence, the range of values of D∗ is dense in Ln2 and consists (see Theorem 1) of all vector functions of the form −ż1 − z1 − z2 1 , z1 ∈ W2 ([t0 , T ]), z1 (T ) = 0, z2 ∈ L2 ([t0 , T ]) . z1 f1 ∈ / R(D∗ ) if f2 ∈ / W12 ([t0 , T ]). Thus, the range of values of D∗ and, hence, R(D) are f2 not closed. Note that the conditions of Theorem 3 are not satisfied because C20 (ε2 E + C40 C4 )−1 = −ε−2 . Let us show that, in this case, a solution of Eq. (5) can also be approximated with the use of solutions of (12) for (f (·), f0 ) ∈ R(D). It is easy to verify that the boundary-value problem (12) takes the form This implies that ẋ1 (t) = x1 (t) + (1 + ε−2 )z1 (t) + f1 (t), ż1 (t) = −z1 (t) + (1 + ε2 )x1 (t) + f2 (t), x1 (t0 ) − z1 (t0 ) = f01 , z1 (T ) = 0, (17) x2 (t) = −ε2 z1 (t, ε). The Riccati equation (13) has the form k̇(t) = 2k(t) + (1 + ε−2 ) − (1 + ε2 )k 2 (t) := U (t, k), k(t0 ) = 1. (18) We set − k := ε2 − √ ε2 + 3ε4 + ε6 , ε2 + ε4 + k := ε2 + √ ε2 + 3ε4 + ε6 . ε2 + ε4 Using the Picard–Lindelöf theorem, we establish that, for a certain ε0 > 0, we have k − < k(t, ε) < k + , 0 < ε < ε0 , t > t0 , (19) which yields k̇(t, ε) > 0, t ≥ t0 , 0 < ε < ε0 , because U (t, k) = (1 + ε2 )(k − k − )(k + − k). Thus, k(t, ε) ≥ k(t0 , ε) > 0 for t ≥ t0 and 0 < ε < ε0 . Let q(·, ε) denote a solution of the equation qtt (t) − 2qt (t) + (1 + ε−2 )(1 + ε2 )q(t) = 0, By direct substitution, we establish that the function t 7→ Rt q(t, ε) = e 2 t0 (1+ε )k(s,ε)ds qt (t0 ) = 1 + ε2 , q(t0 ) = 1. qt (t, ε) satisfies (18). Hence, (1 + ε2 )q(t, ε) > 0 ⇒ qt (t, ε) ≥ 0, t ≥ t0 , 0 < ε < ε0 . Carrying out differentiation, one can easily verify that Zt q(τ, ε) q̇(τ, ε)f2 (τ ) ϕ(t, ε) = f1 (τ ) − τ −t0 dτ f10 + τ −t 0 q(t, ε) e e (1 + ε2 ) et−t0 t0 S. M. Z HUK 484 is a solution of (14) and q(t, ε) z(t, ε) = − t e ZT es f2 (s) + (1 + ε2 )ϕ(s, ε) ds q(s, ε) t is a solution of (15). Therefore, x1 (t, ε) = k(t, ε)z(t, ε) + ϕ(t, ε) and x2 (t, ε) = −ε−2 z(t, ε). If, in addition, f1 (t) ≡ 0, f2 (t) = −et−t0 , and f10 = 1, then the equation Dx(·) = (f (·), f0 ) has the unique solution ! x1 (t) x(t) = , x1 (t) = −f2 (t), x2 (t) ≡ 0 x2 (t) by virtue of the injectivity of D. On the other hand, we have et−t0 ε2 et−t0 + , ϕ(t, ε) = (1 + ε2 )q(t, ε) 1 + ε2 z(t, ε) = q(t, ε) −ε2 t+t0 e ZT e2s ds. q 2 (s, ε) t We show that x1 (·, ε) → x1 and x2 (·, ε) → 0 in Ln2 . Since the function q(·, ε) is increasing, we get et−t0 et−t0 ε2 et−t0 < ϕ(t, ε) ≤ + , 2 2 1+ε 1+ε (1 + ε2 )q(t0 , ε) q(t, ε) z(t, ε) ≤ −ε t+t0 2 e q (t, ε) 2 ZT e2s ds. t Using the equality q(t0 , ε) = 1 and relation (19), we obtain ZT 2 ZT (ϕ(t, ε) + f2 (t)) dt ≤ t0 ε2 et−t0 et−t0 + − et−t0 1 + ε2 (1 + ε2 ) 2 dt → 0, ε → 0, t0 ZT 2 ZT 2 k (t, ε)z (t, ε) dt ≤ t0 2 +e −ε k 2T − e2t 2 2et+t0 dt → 0, ε → 0. t0 Thus, x1 (·, ε) → −f2 (·). One can verify that q(t, ε) = (ε4 + √ √ ε2 + 3ε4 ε6 )e ε2 + ε2 +3ε4 +ε6 (t−t0 ) ε2 + √ 2 ε2 + 3ε4 + ε6 √ √ ε2 − ε2 +3ε4 +ε6 (t−t0 ) ε2 ( ε2 + 3ε4 + ε6 − ε4 )e √ + . 2 ε2 + 3ε4 + ε6 Therefore, the norm of q(·, ε) unboundedly increases as ε → 0. On the other hand, we have −ε−2 z(t, ε) ≤ Therefore, x2 (·, ε) → 0. e2T − e2t . 2et+t0 q(t, ε) C LOSEDNESS AND N ORMAL S OLVABILITY OF AN O PERATOR 485 The next example illustrates the application of sufficient conditions for normal solvability (Theorem 3) to a descriptor equation that is not decomposed into algebraic and differential components. Example 3. We set 1 2 F = 1 − 2 1 − 2 , 1 2 1 − 2 C(t) ≡ 1 2 1 − 2 . 1 2 The corresponding descriptor system has the form d dt 1 1 1 1 x1 − x2 (t) = − x1 (t) − x2 (t) + f1 (t), 2 2 2 2 d dt 1 1 1 1 − x1 + x2 (t) = x1 (t) + x2 (t) + f2 (t), 2 2 2 2 1 1 1 1 0 x1 − x2 (t0 ) = f1 , − x1 + x2 (t0 ) = f20 . 2 2 2 2 We set 1 √2 T := 1 −√ 2 1 √ 2 , 1 √ 2 1 0 Λ := . 0 0 Then F = T ΛT 0 and 0 T C(t)T = 0 0 . 1 0 According to Theorem 3, the corresponding operator D is normally solvable because C2 Q(e) ≡ 0. 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