c Pleiades Publishing, Ltd., 2009. ISSN 0012-2661, Differential Equations, 2009, Vol. 45, No. 3, pp. 416–428. c V.A. Terletskii, E.A. Lutkovskaya, 2009, published in Differentsial’nye Uravneniya, 2009, Vol. 45, No. 3, pp. 403–415. Original Russian Text
PARTIAL DIFFERENTIAL EQUATIONS
Generalized Solution of a Nonlinear Wave Equation with Nonlinear Boundary Conditions of the First, Second, and Third Kinds V. A. Terletskii and E. A. Lutkovskaya Institute for Mathematics, Economics, and Computer Science, Irkutsk State University, Irkutsk, Russia Received October 1, 2007
Abstract— For the wave equation, we construct a generalized solution that has maximum possible analytic properties. We establish a pointwise estimate for the growth of the solution relative to the initial data, prove that certain linear combinations of the first partial derivatives of the solution have integrable derivatives along the characteristics, justify an integration-by-parts formula, and obtain the adjoint of the original differential operator. DOI: 10.1134/S0012266109030112
INTRODUCTION The classical notion of solution of differential problems is essentially restricted in its use in numerous applications as well as in some fields of mathematics itself. This is mainly explained by strict conditions imposed on the smoothness of the functions describing the differential equations and the initial–boundary conditions, or, briefly, on the problem input data. The restrictiveness of the conditions imposed on the problem parameters often not only makes the very notion of classical solution inconvenient but also renders it unusable, which explains why it is so important to construct solutions in the generalized sense. In the present paper, we use the techniques and results in [1, 2]. First, we reduce the original differential problem for the wave equation to a hyperbolic system of four semilinear (a linear differential operator and a nonlinear right-hand side) differential equations by equivalence transformations. Then we rewrite this system in terms of Riemann invariants and, in turn, construct an integral equivalent of the new system. A generalized solution is defined as a solution of the resulting Volterra integral system. Note that a similar approach was used in particular for constructing absolutely continuous solutions of systems of ordinary differential equations [3, 4] and solutions in the “wide sense” (continuous but not smooth) of hyperbolic systems [5]. The main goal of the present paper is to introduce a notion of generalized solution of the wave equation; such a solution should have maximum possible analytic properties and hence be most convenient for the investigation of optimal control problems, where, by virtue of the discontinuity of admissible controls, classical solutions do not exist. For the constructed solution, we obtain a pointwise estimate of growth relative to the input data, prove that certain linear combinations of first partial derivatives of the solution have integrable derivatives along the characteristics, establish an integration-by-parts formula, and obtain the adjoint of the original differential operator. These properties of the constructed solution successfully supplement known results earlier obtained for other generalized solutions of the wave equation (e.g., see [6–8]) and permit one to use the scheme in [9, 10] for the analysis of optimal control problems. 1. STATEMENT OF THE PROBLEM In the rectangle Π = S × T , S = (s0 , s1 ), T = (t0 , t1 ), on the plane of the variables (s, t) with ¯ = Π, ¯ consider the wave equation boundary ∂Π, Π ∪ ∂ Π xtt − a2 (s)xss = f (x, xt , xs , s, t) 416
(1)
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with the initial conditions x(s, t0 ) = x0 (s),
xt (s, t0 ) = x1 (s),
s ∈ S,
(2)
and the boundary conditions xt (s0 , t) = q 0 (x(s0 , t), t),
t ∈ T.
xs (s1 , t) = q 1 (x(s1 , t), t),
(3)
Here x = x(s, t) is the desired solution. We subject the given functions a = a(s),
f = f (x, xt , xs , s, t),
x0 = x0 (s),
x1 = x1 (s),
and q i = q i (x, t), i = 0, 1, to the following restrictions: a is a smooth function on the closed interval S¯ and satisfies the conditions 0 < a0 ≤ a(s) ≤ a∞ < +∞ with some constants a0 and a∞ ; the functions f , q 0 , and q 1 are Lipschitz continuous in the variables x, xt , and xs for almost all fixed values of s and t and pth-power Lebesgue integrable with respect to independent variables s and t for fixed values of x, xt , and xs ; i.e., f (x, xt , xs , · , ·) ∈ Lp (Π), q i (x, ·) ∈ Lp (T ), x, xt , xs ∈ R, and x0 , x0 , x1 ∈ Lp (S). Let us clarify the choice of the structure of the boundary conditions (3). On the left boundary s = s0 of the rectangle Π, the solution x satisfies an ordinary differential equation, which is essentially equivalent to a classical condition of the first kind. On the right boundary s = s1 of the rectangle Π, the second relation in (3) serves any of the two remaining types of boundary conditions simultaneously. If q 1 is independent of x, then relation (3) coincides with a condition of the second kind; if q 1 is a linear function of x, then relation (3) is equivalent to a standard condition of the third kind. Obviously, we have fixed the condition types on the left and right boundaries only to be definite. Indeed, the roles of the boundaries from the viewpoint of the kinds of boundary conditions can change; moreover, all results obtained below are independent of this choice. Note that, in addition to the mentioned generality, conditions (3) has some symmetry and permit readily taking into account boundary controls. It is this goal that we have in mind when assuming the discontinuity of the functions q 0 and q 1 with respect to t. Note also that the differential form of the condition of the first kind on the left boundary automatically guarantees the continuous sewing of the solution at the point (s0 , t0 ). The abovementioned specific features of the form (3) of the boundary conditions show that this form is not only concise but also has more generality than other statements of boundary conditions known to the authors, in particular, those used in [11, 12]. Since the function f is discontinuous in (s, t) ∈ Π and the functions q 0 and q 1 are discontinuous in t ∈ T , it follows that problem (1)–(3) can have solutions only in the generalized sense. In the present paper, we construct such a solution and clarify its properties. 2. REDUCTION OF PROBLEM (1)–(3) TO AN INITIAL–BOUNDARY VALUE PROBLEM FOR A FIRST-ORDER SYSTEM Let x be an arbitrary classical solution of problem (1)–(3). Then its partial derivatives xt and ¯ Let us introduce an xs are continuous and continuously differentiable in the closed domain Π. auxiliary function y = y(s, t) by setting t xs (s, τ ) dτ.
y(s, t) = t0
Then Eq. (1) is equivalent to the system of first-order differential equations t xt − a (s)ys = 2
f [s, τ ] dτ + x1 , t0
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yt − xs = 0.
(4)
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TERLETSKII, LUTKOVSKAYA
Here and in the following, for brevity, by f [s, t] we denote the value f (x(s, t), xt (s, t), xs (s, t), s, t). Along with system (4), we need the system (xt )t − a2 (xs )s = f,
(xs )t − (xt )s = 0
(5)
for the partial derivatives xt and xs . System (5) is primarily needed for the use of the successive approximation method, since in the general case, the partial derivatives xt and xs of the solution x play the same role as x, being arguments of the function f . Moreover, supplementing system (4) with system (5) is reasonable even in the special case in which the function f is independent of the derivatives xt and xs , which makes supplementing system (4) with system (5) when constructing the successive approximation method formally unnecessary. As is shown below, this approach permits one to give the simplest and most precise description of the function class to which the generalized solution x of problem (1)–(3) belongs. Note that system (5) can be viewed as a prolongation (in the terminology of [5, p. 34]) or an extension (in the sense of the definition in [13, p. 72]) of system (4). One can readily see that the initial conditions for the solution of system (4), (5) are given by the relations x(s, t0 ) = x0 (s),
xt (s, t0 ) = x1 (s),
y(s, t0 ) = 0,
xs (s, t0 ) = x0 (s),
s ∈ S,
(6)
and that the relations t x(s0 , t) = x0 (s0 ) +
t q 0 (x(s0 , τ ), τ ) dτ,
t0
xt (s0 , t) = q 0 (x(s0 , t), t),
q 1 (x(s1 , τ ), τ ) dτ,
y(s1 , t) =
(7)
t0
xs (s1 , t) = q 1 (x(s1 , t), t),
t ∈ T,
should be satisfied on the lateral boundaries. 3. INVARIANT FORM OF PROBLEM (4)–(7) Let us write out the matrix function A(s) consisting of the coefficients multiplying the partial derivatives with respect to the space variable s in systems (4) and (5) : 2 0 −a . A= −1 0 The eigenvalues of the matrix A are the functions ±a. Consider the diagonal matrix Λ = diag{−a, a}. Let us construct the matrices 1 a a 1 1 L= , P= 2a 1 −1 a −a consisting of the left and right eigenvectors of A. Obviously, LT P = E, where T stands for transposition and E is the identity matrix. Then [5, p. 28] the Riemann invariants r = (r − , r + ) and the solution x = (x, y) of system (4) are related by the linear nondegenerate transformation r = LT x,
x = Pr.
(8)
By virtue of the second relation in system (4), xs = yt ; therefore, the solution xt = (xt , xs ) of system (5) is related to its Riemann invariants rt = (rt− , rt+ ) by the formulas rt = LT xt ,
xt = Prt .
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To make the formulas shorter, we use the notation x = (x, y, xt , xs ), r = (r − , r + , rt− , rt+ ), r− = (r − , rt− ), and r+ = (r + , rt+ ). By a straightforward verification, one can show that the substitution (8), (9) reduces system (4), (5) to the invariant system rt∓
∓
ars∓
t =
f [s, τ ]dτ + x1 − a (r − − r + )/2,
(rt∓ )t ∓ a(rt∓ )s = f − a (rt− − rt+ )/2;
(10)
t0
the initial conditions (6) give rise to the corresponding conditions r ∓ (s, t0 ) = x0 (s),
rt∓ (s, t0 ) = x1 (s) ± a(s)x0 (s),
(11)
and the boundary conditions (7) give the conditions t
−
+
r (s1 , t) = r (s1 , t) + 2a(s1 ) t0
−
q 1 ((r + (s1 , τ ) + r − (s1 , τ ))/2, τ ) dτ,
t
r (s0 , t) = −r (s0 , t) + 2 x (s0 ) + +
0
0
+
−
q ((r (s0 , τ ) + r (s0 , τ ))/2, τ ) dτ ,
(12)
t0
rt− (s1 , t) = rt+ (s1 , t) + 2a(s1 )q 1 ((r + (s1 , t) + r − (s1 , t))/2, t), rt+ (s0 , t) = −rt− (s0 , t) + 2q 0 ((r + (s0 , t) + r − (s0 , t))/2, t). 4. CHARACTERISTICS AND AN INTEGRAL EQUIVALENT OF THE INVARIANT SYSTEM It is known [5, p. 45; 13, p. 78] that the characteristics of systems (4), (5) and (10) are defined as the integral curves of the ordinary differential equations ds = ±a(s). dt
(13)
By s = s− (ξ, τ ; t) [respectively, s = s+ (ξ, τ ; t)] we denote the solution of the characteristic equation (13) for the negative (respectively, positive) eigenvalue of the matrix A passing through a point ¯ i.e., s± (ξ, τ ; τ ) = ξ. The smoothness of the function a ensures the existence, unique(ξ, τ ) ∈ Π; ness, and smoothness (including smoothness with respect to the initial data) of the functions ¯ t ∈ T¯. s = s± (ξ, τ ; t), (ξ, τ ) ∈ Π, Compared with system (4), (5), the invariant system (10) is specified by a simple structure of its generating differential operators. Each of them can be considered as the derivative of one of the Riemann invariants in the direction specified by a characteristic or as the derivative of a composite function of the arguments (s± (ξ, τ ; t), t) with respect to the variable t. This permits one to construct an integral equivalent of the differential problem in terms of Riemann invariants. Let (ˇ s± (s, t), tˇ± (s, t)) ∈ ∂Π be the initial point of the corresponding characteristic passing through the point (s, t), and let t g0 (r, s, t) = f [s, τ ]dτ + x1 − a (r − − r + )/2, t0
g1 (r, s, t) = f − a (rt− − rt+ )/2,
g = (g0 , g1 ).
Then, by integrating the equations in system (10) along the corresponding characteristics, we obtain the following system of Volterra integral equations equivalent to (10)–(12) : ∓
∓
∓
ˇ∓
t
s (s, t), t (s, t)) + r (s, t) = r (ˇ tˇ∓
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g(r(s, t), s∓ (s, t, τ ), τ ) dτ.
(14)
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TERLETSKII, LUTKOVSKAYA
The validity of the inverse passage from the integral system (14) to the differential system (10) can be proved by a straightforward verification, which heavily relies on the identities ± s± t (s, t; τ ) ± a(s)ss (s, t; τ ) ≡ 0, ± t± t (s, t; τ ) ± a(s)ts (s, t; τ ) ≡ 0,
¯ (s, t) ∈ Π,
τ ∈ T¯,
(15)
¯ ξ ∈ S.
Here the functions τ = t± (s, t; ξ) are the inverse functions of ξ = s± (s, t; τ ) and are solutions of the “reciprocal” equations (13), dτ /dξ = ±1/a(ξ). Note that each of identities (15) implies the independence of an integral curve of the choice of an “initial” point on the same curve and can be obtained by the differentiation of the relations ξ = s± (s± (ξ, τ, t), t; τ ),
τ = t± (s, t± (ξ, τ ; s); ξ),
¯ (ξ, τ ) ∈ Π,
t ∈ T¯,
¯ s ∈ S,
with respect to t and s, respectively. 5. GENERALIZED SOLUTION. EXISTENCE, UNIQUENESS, AND GROWTH ESTIMATES As was shown above, the integral system (14) is equivalent to the differential initial–boundary value problem (10)–(12) in terms of the invariants. Since the smooth solutions of the integral problem (14) are related to those of problem (4)–(7) by the one-to-one transformation (8), (9), we can introduce the following definitions. Definition 1. A generalized solution of the hyperbolic system (4), (5) with the initial–boundary condition (6), (7) is a vector function x related to a solution r of the integral system (14) by (8) and (9). Definition 2. A generalized solution of the wave equation (1) with the initial conditions (2) and the boundary conditions (3) is a function x that is the first coordinate in a generalized solution x of problem (4)–(7). Therefore, the existence and uniqueness conditions and the properties of a generalized solution x of the initial–boundary value problem (1)–(3) are completely determined by the existence and uniqueness conditions and the properties of a solution of the integral problem (14), which, in turn, can naturally be treated as a generalized solution of the differential problem (10)–(12). Let us proceed to the exposition of the successive approximation method, which permits one to prove the existence and uniqueness and obtain growth estimates for a solution of the system of integral equations (14). First, let us construct a rectangle Π ⊆ Π such that at most one of the two characteristics ξ = s− (s, t; τ ) and ξ = s+ (s, t; τ ) issuing from any point (s, t) ∈ Π reaches the lateral boundary. Obviously, Π = S ×(t0 , t ), where t is the root of the equation s+ (s0 , t0 ; t ) = s− (s1 , t0 ; t ). If t ≥ t1 , then we set Π = Π. In turn, we split the rectangle Π into three curvilinear triangles Π10 , Π00 , and Π01 by the rule Π10 = {(ξ, τ ) ∈ Π : s0 < ξ < s+ (s0 , t0 ; τ )}, Π00 = {(ξ, τ ) ∈ Π : s+ (s0 , t0 ; τ ) ≤ ξ ≤ s− (s1 , t0 ; τ )}, Π01 = {(ξ, τ ) ∈ Π : s− (s1 , t0 ; τ ) < ξ < s1 }. Obviously, once we prove the existence and uniqueness of a generalized solution in the domain Π , we can claim its existence and uniqueness in the entire rectangle Π. Indeed, the rectangle Π can be “truncated” to Π by transferring the initial conditions from the boundary t = t0 to the boundary t = t . It suffices to repeat this procedure finitely many times, after which the generalized solution will be extended to the entire rectangle Π. The specific properties of the trajectories of characteristics passing through the points of the domains Π10 , Π00 , and Π01 determine technical differences in the construction of the successive approximation method in each of them. First, we perform this procedure in the domain Π00 . DIFFERENTIAL EQUATIONS
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Let (s, t) ∈ Π00 be an arbitrary fixed point, and let ξ± = s± (s, t; t0 ). The domain of the solution x at the point (s, t) has the form G(s, t) = {(ξ, τ ) ∈ Π00 : s+ (s, t; τ ) < ξ < s− (s, t; τ ), τ ∈ (t0 , t)}. We define the sequence of approximations {r (k) , rt(k) }, k = 0, 1, . . . , in accordance with system (14) by the relations r ±(k+1) (s, t) = r ± (ξ± , t0 ) +
t τ f (k) [ξ, α] dα + x1 (ξ) t0
t0
r −(k) (ξ, τ ) − r +(k) (ξ, τ ) − a (ξ) 2 t
±(k+1)
rt
(s, t) = rt± (ξ± , t0 ) + − a (ξ)
dτ,
ξ=s± (s,t;τ )
(16)
f (k) [ξ, τ ] t0
−(k) rt (ξ, τ )
− rt 2
+(k)
(ξ, τ )
dτ,
ξ=s± (s,t;τ )
∓(0)
are set to zero, the symbol f (k) [ξ, α] is treated as the where the initial functions r ∓(0) and rt (k) value of the function f evaluated at the point (ξ, τ ) on the kth approximation x(k) , x(k) t , xs , and ± ± the initial values r (ξ± , t0 ) and rt (ξ± , t0 ) are specified by conditions (11). (k)
We introduce the norms δ(k) and δt
of the increments by setting
δ(k) (ξ, τ ) = r (k+1) (ξ, τ ) − r (k) (ξ, τ ) ,
δt (ξ, τ ) = rt (k)
(k+1)
(ξ, τ ) − rt (ξ, τ ) (k)
for k = 0, 1, . . . and (ξ, τ ) ∈ Π. Then, by virtue of (16) and the Lipschitz condition for the function f , we have the inequalities t τ δ(k) (s, t) ≤ K t0
t +
(k−1) (δ(k−1) (ξ, α) + δt (ξ, α)) dα + δ(k−1) (ξ, τ )
t0
(k−1) (δ(k−1) (ξ, α) + δt (ξ, α)) dα + δ(k−1) (ξ, τ )
t0 (k) δt (s, t)
≤K
t
(k−1)
(δ(k−1) (ξ, τ ) + δt
t0 (k−1)
+ (δ(k−1) (ξ, τ ) + δt
(ξ, τ ))
(ξ, τ ))
ξ=s− (s,t;τ )
dτ, ξ=s+ (s,t;τ )
(17)
ξ=s− (s,t;τ )
ξ=s+ (s,t;τ )
dτ.
Here k = 1, 2, . . . , and the constant K can be computed in terms of the Lipschitz constant L of the function f by the formula √ K = ( 2L max{1, 1/a0 } + max |a (s)|)/2. ¯ s∈S
Note that both double integrals in the first inequality in (17) have the integration domain G(s, t). In each of these integrals, we make the corresponding change of variables ξ = s± (s, t; τ ) and obtain DIFFERENTIAL EQUATIONS
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TERLETSKII, LUTKOVSKAYA
the chain of relations t τ (δ(k−1) (ξ, α) + δt(k−1) (ξ, α))|ξ=s− (s,t;τ ) dα t0
t0
τ (δ
+
(k−1)
(ξ, α) +
(k−1) δt (ξ, α))|ξ=s+ (s,t;τ )
dα dτ
t0
t
s
(k−1)
δ(k−1) (ξ, α) + δt a(ξ)
= t0
(ξ, α)
dξ −
(k−1)
δ(k−1) (ξ, α) + δt a(ξ)
(ξ, α)
dξ dα
s− (s,t;α)
s+ (s,t;α)
=
s
(k−1)
δ(k−1) (ξ, α) + δt a(ξ)
(ξ, α)
dξ dτ ≡ I (k−1) (s, t).
G(s,t)
Then the first inequality in (17) can be represented in the form
t δ(k) (s, t) ≤ K I (k−1) (s, t) + [δ(k−1) (s− (s, t; τ ), τ ) + δ(k−1) (s+ (s, t; τ ), τ )] dτ .
(18)
t0
This, together with the second inequality in (17) for k = 1, implies that
t δ(1) (s, t) ≤ K I (0) (s, t) + [δ(0) (s− (s, t; τ ), τ ) + δ(0) (s+ (s, t; τ ), τ )] dτ , t0
(19)
t δt (s, t) ≤ K (1)
(0)
(0)
[(δ(0) (ξ, τ ) + δt (ξ, τ ))|ξ=s− (s,t;τ ) + (δ(0) (ξ, τ ) + δt (ξ, τ ))|ξ=s+ (s,t;τ ) ] dτ. t0
Note that
δ(0) (ξ, τ ) = r (1) (ξ, τ ) ≤ |r + (s+ (ξ, τ ; t0 ), t0 )| + |r − (s− (ξ, τ ; t0 ), t0 )| τ α (0) 1 |f [η, β]| dβ + |x (η)| + + t0
α
+
η=s (ξ,τ ;α)
t0
|f (0) [η, β]| dβ + |x1 (η)|
dτ
η=s− (ξ,τ ;α)
t0 +
(20)
= |r (s (ξ, τ ; t0 ), t0 )| + |r − (s− (ξ, τ ; t0 ), t0 )| +
+
|f (0) [η, α]| dη dα + a(η)
=
(1) rt (ξ, τ ) τ
|x1 (η)| dη, a(η)
s+ (ξ,τ ;t0 )
G(ξ,τ ) (0) δt (ξ, τ )
s− (ξ,τ ;t0 )
≤
|rt+ (s+ (ξ, τ ; t0 ), t0 )|
+ |rt− (s− (ξ, τ ; t0 ), t0 )|
(|f (0) [s+ (ξ, τ ; α), α]| + |f (0) [s− (ξ, τ ; α), α]|) dα.
+ t0
By strengthening inequalities (19) with the use of these estimates, we obtain two types of terms on the right-hand side of the new estimate; these terms contain either the initial data or the right-hand side of Eq. (1). Each type occurs in its own integral. DIFFERENTIAL EQUATIONS
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Let us construct the corresponding estimates. For brevity, we consider some abstract function ϕ = ϕ(s); this notation combines the absolute values of the coordinates of the vector functions r = r(s, t0 ) and rt = rt (s, t0 ). Obviously, if the relation ξ = s± (s, t; τ ) holds for (ξ, τ ) ∈ Π00 , then s± (ξ, τ ; t0 ) = ξ± . Consequently, t
ϕ(s± (s± (s, t; τ ), τ ; t0 )) dτ = ϕ(ξ± )(t − t0 ).
t0
A similar integral for characteristics in different families can be computed in a slightly more complicated way. Here we replace the variable τ by a new variable η by setting η = s± (s∓ (s, t; τ ), τ ; t0 ). Then, by taking into account Eq. (13) and the first identity in (15), we obtain ∓ dη = ∓2a(s∓ (s, t; τ ))s± ξ (s (s, t; τ ), τ ; t0 ) dτ.
One can readily show that the derivative s± ξ (ξ, τ ; t) satisfies the relations t t ± da(s (ξ, τ ; α)) a (s± (ξ, τ ; α)) dα = exp s± ξ (ξ, τ ; t) = exp ± a(s± (ξ, τ ; α)) τ
t = exp
τ
d ln a(s± (ξ, τ ; α))
=
a(s± (ξ, τ ; t)) . a(ξ)
τ
In particular, this readily implies the two-sided estimate a∞ a0 ≤ s± ξ (ξ, τ ; t) ≤ a∞ a0
(21)
and the relation dη = ±2a(s± (ξ, τ ; t)) dτ . Therefore, t
ϕ(s± (s∓ (s, t; τ ), τ ; t0 )) dτ =
t0
1 2
ξ
−
ϕ(η) dη. a(η)
(22)
ξ+
In addition to one-dimensional integrals, the initial data also occur in the integral over the domain G(s, t). In this case, after the replacement of the variable ξ by the variable η = s± (ξ, τ ; t0 ) and the use of inequality (21), we obtain
t ξ−
±
ϕ(s (ξ, τ ; t0 )) dξ dτ =
ϕ(η)s± η (η, t0 ; τ ) dη
a∞ dτ ≤ a0
t0 ξ+
G(s,t)
=
t ξ− ϕ(η) dη dτ t0 ξ+
a∞ (t − t0 ) a0
ξ− ϕ(η) dη. ξ+
Finally, one can readily see that t
(|x1 (s+ (s, t; τ ))| + |x1 (s− (s, t; τ ))|) dτ =
t0
DIFFERENTIAL EQUATIONS
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|x1 (ξ)| dξ. a(ξ)
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TERLETSKII, LUTKOVSKAYA
Now consider the terms that contain the function f (0) as the integrand. In the integral over the domain G(s, t), we replace the variable ξ by the variable η = s± (ξ, τ ; α). Then, by taking into account (21), we obtain τ
a∞ |f (0) [s± (ξ, τ ; α), α]| dα dξ dτ ≤ a0
G(s,t) t0
t τ
s− (s,t;α)
|f (0) [η, α]| dη dα dτ t0 t0 s+ (s,t;α)
a∞ (t − t0 ) ≤ a0
|f (0) [ξ, τ ]| dξ dτ. G(s,t)
The double integrals containing the function f (0) as the integrand can be of two types, repeated integrals over the characteristics ξ = s± (s, t; τ ) and double integrals over the domain G(s, t). In the first case, by virtue of the obvious identity s± (s± (s, t; τ ), τ ; α) = s± (s, t; α), the repeated integral is computed by the classical change of the order of integration, tτ |f
(0)
±
tt
±
[s (s (s, t; τ ), τ ; α), α]| dα dτ =
t0 t0
|f (0) [s± (s, t; α), α]| dτ dα
t0 α
t =
(t − τ )|f (0) [s± (s, t; τ ), τ ]| dτ.
t0
In the second case, one should replace the variable τ by the variable η = s± (s∓ (s, t; τ ), τ ; α) as was done in the derivation of formula (22) but with a small difference; namely, we use the constant t0 instead of the variable α in the last relation. We omit the transformations, which are virtually the same, and only write out the definitive result, tτ
|f (0) [s± (s∓ (s, t; τ ), τ ; α), α]| dα dτ =
t0 t0
1 2
|f (0) [ξ, τ ]| dξ dτ. a(ξ)
G(s,t)
We have listed all cases of terms (and methods for estimating them from above) that appear in the strengthening of inequalities (19) with the use of (20). After the above-mentioned manipulations, we obtain the following estimates for the norms of the differences δ(1) and δt(1) via the input data: ξ− δ(1) (ξ, τ ) ≤ C |x0 (ξ− )| + |x0 (ξ+ )| +
(|x0 (ξ)| + |x1 (ξ)| + |x0 (ξ)|) dξ ξ+
|f (0) [ξ, τ ]| dξ dτ ,
+ G(s,t)
(1) δt (ξ, τ )
ξ− ≤ C |x (ξ− )| + |x (ξ+ )| + |x (ξ− )| + |x (ξ+ )| + (|x0 (ξ)| + |x1 (ξ)| 1
1
0
0
(23)
ξ+
t + |x0 (ξ)|) dξ +
t0
(|f (0) [s+ (s, t; τ ), τ ]| + |f (0) [s− (s, t; τ ), τ ]|) dτ
|f (0) [ξ, τ ]| dξ dτ .
+ G(s,t)
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Here the constant C is independent of the input data f , x0 , and x1 of the problem and is solely determined by the values of the constants K, a0 , and a∞ . (2) Since the functions δ(2) and δt satisfy inequalities (18) and the second inequality in (17) for (1) k = 2, it follows that they can be estimated by analogy with the functions δ(1) and δt by inequalities of the form (23), where the constant C differs only by the coefficient t − t0 . Obviously, the form of (3) the estimate is also preserved for the functions δ(3) and δt but contains the new factor (t − t0 )2 /2! multiplying the constant C. Therefore, if we denote the expressions in curly braces in the first and second inequality in (23) by D(s, t) and Dt (s, t), respectively, then we have ∞
δ(k) (s, t) ≤ eC(t−t0 ) D(s, t),
k=0
∞
δt(k) (s, t) ≤ eC(t−t0 ) Dt (s, t).
k=0
Therefore, the series ∞ (r ±(k+1) (s, t) − r ±(k) (s, t)),
∞
k=0
k=0
±(k+1)
(rt
±(k)
(s, t) − rt
(s, t)) ±(k)
are absolutely convergent, and consequently, the sequences {r ±(k) (s, t)} and {rt (s, t)}, k = 0, 1, . . . , have limits almost everywhere in Π00 ; these limits will be denoted by r ± (s, t) and rt± (s, t), respectively. We have thereby shown that there exists a unique generalized solution r = (r − , r + , rt− , rt+ ) of problem (10)–(12). Consequently, there exists a unique generalized solution x of the original problem (1)–(3). In addition, this solution and its partial derivatives satisfy the growth estimates (xt (s, t), xs (s, t)) ≤ M Dt (s, t)
|x(s, t)| ≤ M D(s, t),
(24)
with some constant M independent of the specific choice of the input data of problem (1)–(3). In a complete analogy with the above-mentioned considerations, one can construct the generalized solution in the remaining triangles Π10 and Π01 . The differences in the resulting estimate (24) are the following. First, the dependence domain G(s, t) for the triangles Π10 and Π01 is given by the relations G(s, t) = {(ξ, τ ) ∈ Π : max{s0 , s+ (s, t; τ )} < ξ < s− (s, t; τ ), τ ∈ (t0 , t)}, G(s, t) = {(ξ, τ ) ∈ Π : s− (s, t; τ ) < ξ < min{s+ (s, t; τ ), s1 }, τ ∈ (t0 , t)}, respectively. Second, the expressions D(s, t) and Dt (s, t) change. In the case of the domain Π10 , one should set ξ+ = s0 in these expressions and add the integral of the function |q 0 (0, t)| with respect to t ∈ (t0 , tˇ+ (s, t)) to D(s, t) and the term |q 0 (0, tˇ+ (s, t))| to Dt (s, t). For the triangle Π01 , one should set ξ− = s1 and add the integral of the function |q 1 (0, t)| over the interval t ∈ (t0 , tˇ− (s, t)) to D(s, t) and the term |q 1 (0, t)| to Dt (s, t). 6. PROPERTIES OF THE GENERALIZED SOLUTION Let us clarify the analytic properties of the generalized solution and describe a function class to which it belongs for arbitrary input parameters of the problem satisfying the assumptions of Section 1. It readily follows from the above-represented proof of the existence and uniqueness of a generalized solution that all Riemann invariants r ± and rt± are Lebesgue integrable functions on Π. Indeed, by using the inequality n p n bi ≤ np−1 bpi , bi ≥ 0, i = 1, 2, . . . , n, i=1
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the H¨ older inequality, and the estimates (24), we obtain the relation ( r(s, t) p + rt (s, t) p ) ds dt
Π
|f
≤ C1
(0)
(|x0 (s)|p + |x0 (s)|p + |x1 (s)|p ) ds
p
[s, t]| ds dt +
Π
S
(|q (0, t)| + |q (0, t)| ) dt , 0
+
p
1
p
T
where C1 is a constant independent of the functions occurring in brackets and determined only by the constant C and the size of the domain Π. By virtue of the Lebesgue p-integrability of the input data, the integrals on the right-hand side of the inequality exist and are finite; therefore, r, rt ∈ Lp (Π). In fact, the Riemann invariants belong to a narrower function space. Indeed, since the functions rt± satisfy the integral relations (10), it follows that they are absolutely continuous along their characteristics. In particular, this implies the existence of traces of the Riemann invariants rt± on an arbitrary rectifiable curve in the rectangle Π provided that the curve has no common parts of nonzero measure with the corresponding characteristic. In addition, the derivatives along the corresponding characteristics are well defined for them almost everywhere in Π. Set
d ± ± d ± (r (s, t))± = r (s (s, t; τ ), τ ) . (25) dt dτ τ =t Then the solutions rt of the integral system (14) satisfy the differential equations + − drt (s, t) drt (s, t) = = f − aa xs dt dt − +
d ± r almost everywhere in Π, and the derivatives themselves satisfy the inclusions ∈ Lp (Π). dt t ± We have thereby shown that, for arbitrary input data f , x0 , x1 , q 0 , and q 1 of problem (1)–(3) satisfying the assumptions listed in Section 1, the solution r of the invariant problem (10)–(12) belongs to the function space in which the norm is given by the expression d + d − r r + . r Lp (Π) + rt Lp (Π) + dt t + L (Π) dt t − L (Π) p
p
Now let us return to the generalized solution x of problem (1)–(3). This solution and its derivatives xt and xs are related to the generalized solution r, rt of problem (10)–(12) by the linear nondegenerate transformation (8), (9). Therefore, x belongs to the function space with the norm d d + . x Lp (Π) + xt Lp (Π) + xs Lp (Π) + dt (xt − axs )+ dt (xt + axs )− Lp (Π) Lp (Π) p1 (Π). We denote this space by W 7. FORMULAS FOR INTEGRATION BY PARTS AND THE ADJOINT DIFFERENTIAL OPERATOR Although in general the second partial derivatives xtt , xst , and xss of the generalized solution x do not exist even in the generalized sense, one can extend the domain of the differential operator DIFFERENTIAL EQUATIONS
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1 (Π). Indeed, Dx = xtt −a2 xss from the space of twice differentiable functions x in Π to the space W p 2 let x ∈ C (Π). Then, by (9) and (25), we have d ± (r )± = (rt± )t ± a(rt± )s = (xt ∓ axs )t ± a(xt ∓ axs )s dt t = xtt ∓ axst ± axts − a2 xss − aa xs = Dx − aa xs . Therefore, d Dx = dt
d (x)± dt
+ a axs
(26)
∓
for arbitrary x ∈ C 2 (Π). In addition, as was mentioned above, the expression on the right-hand p1 (Π). side exists almost everywhere in Π and belongs to the space Lp (Π) provided that x ∈ W 1 (Π), and they can be Consequently, formulas (26) are well defined for arbitrary functions x ∈ W p 1 (Π). treated as a definition of the operator D on the space W p The absolute continuity of the Riemann invariants rt± along the corresponding characteristics ξ = s± (s, t; τ ) permits one to justify a formula for integration by parts despite the discontinuity of these functions in the domain Π. Let us show that 2 2 ψDx ds dt = [(ψxt − ψt x)ν0 − (ψa xs − (a ψ)s x)ν] dω − xD∗ ψ ds dt (27) Q
∂Q
Q
q (Π), 1/p + 1/q = 1. Here Q is an arbitrary connected p (Π) and ψ ∈ W for arbitrary functions x ∈ W subdomain of the rectangle Π with piecewise √ smooth boundary ∂Q, (ν0 , ν) is the unit outward normal at points of the boundary ∂Q, dω = ds2 + dt2 is the length differential of the boundary, and D ∗ is the adjoint of D given by the formula d∗ d∗ ∗ (ψ)± − (a aψ)s (28) D ψ= dt dt ∓ 1 (Π), where for ψ ∈ W q
d∗ (ψ)± = ψt ± (aψ)s , dt
and by the formula
D ∗ ψ = ψtt − (a2 ψ)ss
(29)
for ψ ∈ C (Π). First, note that the integration-by-parts rule [1, 2] ∗ dz d ψ ψ ds dt = ψz(ν0 ± aν) dω − z ds dt dt ± dt ± 2
Q
Q
∂Q
holds for arbitrary functions z and ψ satisfying the conditions dψ dz ∈ Lp (Π), ψ, ∈ Lq (Π). z, dt ± dt ± dx here and by taking into account the representation (26) of the operator Dx By setting z = dt ∓ and notation (25), we obtain
∗ dx d ψ dx ψDx ds dt = ψ (ν0 ± aν) dω − − ψa axs ds dt dt ∓ dt ± dt ∓ Q Q ∂Q [(ψt ± (aψ)s )xt − ((a2 ψ)s ± aψt )xs ] ds dt. = ψ(xt ∓ axs )(ν0 ± aν) dω − Q
∂Q
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We temporarily suppose that ψ ∈ C 2 (Π). Then the standard integration-by-parts formula can be applied to the integral over the domain Q. By making simple transformations and by using the obvious identity [(ψax)s ν0 − (ψax)t ν] dω ≡ 0, ∂Q
we obtain 2 2 ψDx ds dt = [(ψxt − ψt x)ν0 − (a ψxs − (a ψ)s x)ν] dω − [ψtt − (a2 ψ)ss ]x ds dt. Q
(30)
Q
∂Q
1 (Π) by analogy with the definition We define the operator D ∗ by relation (28) for all functions in W q p1 (Π). Then, by virtue of (29), relation (30) coincides with of the operator D on the space W formula (27), which completes the proof. ACKNOWLEDGMENTS The research was supported by the Russian Foundation for Basic Research (projects nos. 08-0100709-a and 08-01-98007-r-sibir-a). REFERENCES 1. Terletskii, V.A., Generalized Solution of Hyperbolic Systems of One-Dimensional Semilinear Differential Equations, Optimiz. Control , Irkutsk, 2004, vol. 11. 2. Terletskii, V.A., A Generalized Solutions of One-Dimensional Semilinear Hyperbolic Systems with Mixed Conditions, Izv. Vyssh. Uchebn. Zaved. Mat., 2004, no. 12, pp. 75–83. 3. Coddington, E. and Levinson, N., Theory of Ordinary Differential Equations, New York, 1955. Translated under the title Teoriya obyknovennykh differentsial’nykh uravnenii, Moscow: Inostrannaya Literatura, 1958. 4. Sansone, G., Equazioni differenziali nel campo reale. Pt. 1 , Bologna, 1948. Translated under the title Obyknovennye differentsial’nye uravneniya. T.2 , Moscow: Inostrannay Literatura, 1954. 5. Rozhdestvenskii, B.L. and Yanenko, N.A., Sistemy kvazilineinykh uravnenii i ikh prilozheniya k gazovoi dinamike (Systems of Quasilinear Equations and Their Applications to Gas Dynamics), Moscow: Nauka, 1978. 6. Il’in, V.A., Boundary Control of Oscillations on Two Ends in Terms of the Generalized Solution of the Wave Equation with Finite Energy, Differ. Uravn., 2000, vol. 36, no. 11, pp. 1513–1528. 7. Il’in, V.A., Boundary Control of Oscillations at One End and the Other End Fixed in Terms of the Generalized Solution of the Wave Equation with Finite Energy, Differ. Uravn., 2000, vol. 36, no. 12, pp. 1670–1686. 8. Znamenskaya, L.N., Generalized Solutions in the Class L2 of the Second Boundary Value Problem for the Wave Equation, Differ. Uravn., 2004, vol. 40, no. 4, pp. 539–546. 9. Vasil’ev, O.V., Srochko, V.A., and Terletskii, V.A., Metody optimizatsii i ikh prilozheniya. Ch. 2. Optimal’noe upravlenie (Optimization Methods and Their Application. Part 2. Optimal Control), Novosibirsk: Nauka, 1990. 10. Vasil’ev, O.V. and Terletskii, V.A., Iterative Processes, Based on Pontryagin’s Maximum Principle, for Solving Optimal Control Problems in Systems with Lumped and Distributed Parameters, in Optimizatsiya: Modeli. Metody. Resheniya (Optimization: Models, Methods, Solutions), Novosibirsk: Nauka, 1992, pp. 35–54. 11. Terletskii, V.A., Method of Increments in Optimal Control Problem for a Nonlinear Wave Equation, Tr. mezhdunar. konf. “Problemy upravleniya i prilozheniya (Tekhnika, proizvodstvo, ekonomika)”. T. 2. Upravlenie i optimizatsiya (Proc. Int. Conf. “Control Problems and Applications (Technology, Manufacturing, Economics).” Vol. 2. Control and Optimization), Minsk, 2005, pp. 150–158. 12. Terletskii, V.A. and Sokol’nikova, Yu.I., Justification of the Increment Method in Optimal Control Problem for a Nonlinear Wave Equation, Vestn. Buryat. Univ. Ser. 13 Mat., Inform., 2005, vol. 2, pp. 156–166. 13. Godunov, S.K., Uravneniya matematicheskoi fiziki (Equations of Mathematical Physics), Moscow: Nauka, 1979.
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