Optimal control of solutions to the multipoint initial-final problem for nonstationary relatively bounded equations of Sobolev type

Suppose that X , Y , and U are Hilbert spaces, and then take bounded linear operators L E L( X ; Y ) and B E L( U ; Y ) , assuming that the kernel of L is non-trivial. Take also a closed linear operator M E C l( X ; Y ) whose domain is dense in X .

Consider the Sobolev-type equation [1–4]

Lx(t) = a(t)Mx(t) + f (t) + Bu(t)

with a control vector function u : [0, T] ^ U , a vector function f : [0, T] ^ Y of exterior force, and a scalar function a : [0, T ] ^ R ₊ , to be specified later, characterizing the change in time of the parameters of (1). The operators L and M generate the analytic resolving group for the homogeneous stationary equation (1), which means that a(t) = 1 .

We consider an optimal control problem for (1). Namely, we aim to find a pair (x, u) E X × U ad with

J (x, u) =    inf     J (x,u).

( x,u ) e X x U ad

Here U _ad is a closed convex set of admissible controls in the Hilbert space U of controls, all pairs (x,u) satisfy the multipoint initial-final problem [5] for (1), and J (x,u) is a certain penalty functional in special form.

Previously the authors studied the optimal control problem for solutions to non-stationary Sobolev-type equations (1) with the Showalter–Sidorov condition [6, 7]. In this paper we study the optimal control of solutions to the multipoint initial–final problem [5], which is a generalized Showalter–Sidorov problem [8] for (1).

• 1.    Relatively Spectrally Bounded Operators

Recall the standard notation of the theory of relatively p -bounded operators [3].

Starting with two Hilbert spaces X and Y , take a bounded linear operator L E L( X ; Y ) with non-trivial kernel and a closed linear operator M E C l( X ; Y ) whose domain is dense in X . Consider the stationary equation

Lx(t) = Mx(t) + f (t),

called a Sobolev-type equation [3].

Definition 1. The sets p ^L (M ) = {p E C : (pL — M ) ^-1 E L( Y ; X )} and a ^L (M ) = C \ p ^L (M ) are called the L -resolvent set and the L -spectrum of M respectively.

Definition 2. The operator-valued functions (pL — M ) ^-1 , R " (M ) = (pL — M ) ^-1 L , and L " (M ) = L(pL — M ) ^-1 are respectively called the resolvent, right resolvent, and left resolvent of M with respect to L (or briefly the L -resolvent , right L -resolvent , and left L -resolvent of M ).

Lemma 1. Given L E L( X ; Y ) and M E Cl( X ; Y ) , the L -resolvent, right and left L -resolvents of M are analytic on p^L (M ) .

Definition 3. An operator M is called spectrally bounded with respect to an operator L (or briefly (L, a) -bounded) whenever    3r ₀ > 0 Vp E C (|p| > r ₀ ) ^ (p E p L (M )).

Put y = {Ц E C : |p| = r >  r ₀ } . The Riesz-type integrals

P =

P [ R L (M ) dp, 2ni J₇ ^

Q = P [ LL(M) dp 2ni J7 M exist by Lemma 1 for every (L, a)-bounded operator M. The operators P E L(X) and Q E L(Y) are projections [3]. Put X0 = kerP, Y0 = kerQ; X1 = imP, and Y1 = imQ. Denote the restriction of L (M) to Xk by Lk (Mk) for k = 0,1.

Theorem 1. The following claims hold for every (L, a) -bounded operator M :

• (i)    the operators L k , M k : X k ^ Y k for k = 0,1 ;

• (ii)    the operators M ₀ E L( X ⁰ ; Y ⁰ ) and M 1 E Cl( X 1 ; Y 1 ) ;

• (iii)    there exists operators L^-1 E L( Y 1 ; X ¹ ) and M 0¹ E L( Y 0 ; X ⁰ ) ;

• (iv)    there exist analytic resolving operator groups {X t E L( X ) : t E R} for the homogeneous equation (3) and {Y t E L( Y ) : t E R} for the equation R e (M)y/(t) = M(eL — M) ^-1 y(t), where вEp ^L (M ) , which are of the form

X t = e tL i 1 M 1 p = X [ r L ( m )e ^t dp    Y t = e ^{tM 1} L i 1 Q =

JLKMie ^t d^. ^γ

2ni J    ^' '

γ

Theorem 1 implies the existence of the operators H = M ₀ 1 L 0 E L( X ⁰ ) and S = L ₁ ¹ M 1 E L ( X ¹ ) .

Definition 4. An (L, a) -bounded operator M is called

• (i)    (L, 0) -bounded whenever the point to is a removable singularity of the L -resolvent of M , that is, H = O ;

• (ii)    (L,p) -bounded whenever the point to is an order p E N pole of the L -resolvent of M , that is, H ^p = O and H ^p+1 = O ;

• (iii)    (L, to) -bounded whenever the point to is an essential singularity of the L -resolvent of M , that is, H q = O for all q E N .

• 2.    Strong Solutions of the Multipoint Problem

Take two Hilbert spaces X and Y. For two operators L E L(X; Y) and M E Cl(X; Y), where M is (L,p)-bounded for p E {0} U N, introduce the condition aL(M) = J aL(M), n E N, and aL(M) = 0, there is a closed loop Yj C C

j=0                                                     ___ > and Yj = dDj, where Dj D aL(M), such that Dj П ^L(M) = 0

and D k П D l = 0 for all j,k,l = 1,n,k = l.

Define the operators P j E L( X ) and Q j E L( Y ) for j = j, n as

/ L L (M W, j = 1,n γ

P j =       R ^LL (MW, Q j =

2ni LM

γj thanks to the relative spectral theorem [9], and moreover, the results of [9], and the operators    Po = P — £ Pj, Qo = Q — £ Qj.

j=1

Consider the multipoint initial-final problem

Pj(x(t) — Xj) = 0,   (Tj < Tj+i) j = 0,n(5)

for (3). Applying to (3) the projections I — Q and Q j for j = 0,n yields the equivalent system

Hx ⁰ = x ⁰ + M^f ⁰ , X j = S i j x j + L - f ¹

where H = M ^—1 L ₀ E L( X ⁰ ) is a degree p E {0} U N nilpotent operator, the operator S 1j = Lph E Cl( X 1 ) has the range a(S j ) = a L (M ) , while f ⁰ = (I — Q)f , f j ¹ = Q j f , x ⁰ = (I — P )x , and x j = P j x for j = 0, n .

Put N ₀ = {0} U N and construct the space

Hp+1(Y) = {{ e L2(0,t; Y) : ^lp+1> E L2(0,T; Y), p E N} which is a Hilbert space with the inner product

K.n) = L /(^ ⁽” ,4 ⁽” ) y q =0 ₀

dt.

Definition 5. A vector-valued function x E H ¹ ( X ) is called a strong solution to the multipoint initial-final proble m ( 3), (5) whenever it satisfies (3) and the terms of P j (x(T j ) — x j ) = 0 for j = 0,n almost everywhere.

Lemma 2. If an operator M is (L,p) -bounded, with p E N o , then for every vector function f ⁰ E H ^p+1 ( Y ⁰ ) there exists a unique solution X ⁰ E H ¹ ( X ⁰ ) to (6):

pq

■ (t) = - ^H q M o-¹ — f ⁰ (t).

q=0

Lemma 3. Under the assumptions of Lemma 2 , if condition (4) is fulfilled then for every vector X j E X and every vector function f j ¹ E H ( Y } ) there exists a unique solution x j E

τ j

— / XL ///(»>■

t

H ^X ^ ) to the problem P j (x(T j ) — x j ) = 0 for (7): x j (t) = X j ^{T j} x _Tj

Theorem 2. Given vectors x j E X for j = 0, n and a vector function f : [0, т] ^ Y satisfying the assumptions of Lemmas 2 and 3, there exists a unique solution x E H 1 ( X ) :

τ j

— J Xj-LjjwA .

t

p  qn

• ■(t) =—E ^{H q M} -   f ⁰(t)+EI X T x t ,

• 3.    Optimal Control of the Multipoint Problem

q=0                     j=0 \

For a Hilbert space X consider the equation

LX(t) = a(t)Mx(t) + f (t) + Bu(t)

with operators L E L( X ; Y ) , M E C l( X ; Y ) , and B E L( U ; Y ) , a scalar function a : [0, т) ^ R ₊ , as well as vector functions u : [0,т) ^ U and f : [0, т) ^ Y to be specified later.

Take a Hilbert space Z and an operator C E L( X ; Z ) . Consider the penalty functional

1 r

J(u) = E / »z '” q = ⁰ 0

—

kτ z^W^dt + L / (Nu(q),u(q)jUdt,    z = Cx, (9)

q = ⁰ 0

where 0 <  k <  p + 1 . The operators N _q E L( U ) for q = 0,1,...,p + 1 are self-adjoint and positive definite, while z d = z d ( t, s ) is an observation from some space of observations Z . Note that if x E H 1 ( X ) then z E H 1 ( 3 ) . By analogy with H ^p+1 ( Y ) , define the space H ^p+1 ( U ) , which is a Hilbert space because so is U . We distinguish a convex and closed subset H p +1 ( U ) of the space H ^p+1 ( U ) , called the set of admissible controls.

Definition 6. A vector function v E H¹ pd_l^l ( U ) is called an optimal control of solutions to problem (5), (8) whenever

J (v) =       min        J (u),

( x ( u ) ,u ) G X x H ^{p +1} ( U )

where the pairs (x(u),u) E X x H p +1 ( U ) satisfy (5), (8).

By Theorem 2, a unique solution x E H 1 ( X ) to problem (5), (8) exists for all vectors x j E X for j = 0,n , vector functions f E H ^p+1 ( Y ) , u E H ^p+1 ( U ) and a function a E C ^p+1 ([0,T); R ₊ ) separated from zero:

x(t) = — ^L ^H q M 0 ^-1 (I—Q) ( alt) dd )‘ ^{f ( t )} +( B^{u ( t )} + q=0 \ ⁽ ) / ⁽ )

n

+ E I ^V ( A E j=0 \

τ j

- j x At -^ s^ - Q j (f (s) ₊ Bu(s))ds

t

t by analogy with [6]. Here A(t) = j a(q)dq. We now fix xj G X for j = 0, n and f G 0

H ^p+1 ( Y ) and consider (11) as a mapping D : u ^ x(u) .

Lemma 4. Given Hilbert spaces X, Y , and U , take an (L,p) -bounded operator M , with p G N ₀ , a function a G C ^p+1 (R ₊ ; R ₊ ) separated from zero, and fix vectors x j G X for j = 0,n and f G H ^p+1 ( Y ) . Then the mapping D : H ^p+1 ( U ) ^ H ¹ ( X ) defined by (11) is continuous.

Proof. Since B G L(H ^p+1 ( U ); H ^p+1 ( Y )) and (11) is the solution to (8), this lemma holds by the properties of the operator group X t and the continuity of a(t) for t G R ₊ , by analogy with the proof of Theorem 2.

□ Theorem 3. Take an (L,p) -bounded operator M with (p G N ₀ ) and a function a G C ^p+1 ([0 ,t ); R ₊ ) separated from zero. Then for all vectors x j G X for j = 0,n , f G H ^p+1 ( Y ) , and z d G Z, there exists a unique solution v G H p +^^ U ) to the optimal control problem (5), (8)–(10) .

Proof. Using the mapping D of Lemma 4, we see that the functional (9) becomes

J(u) = ||Cx(t; u) - zd^Hi(z) + [n,u], where n(k)(t) = Nku(k^ for k = 0,... ,p + 1. Therefore,

J(u) = n(u,u) - 26(u) + |zd - Cx(t;0)|Hi(Z), where   n(u, u) = |C(x(t; u) — x(t;0))|H 1(Z) + [n,u]   is a coercive continuous bilinear form on Hp+1(U), and

6(u) = (z d - Cx(t; 0), C(x(t; u) - x(t; 0))) _H i ( Z )

is a continuous linear form on H ^p+1 ( U ) . Thus, the theorem is valid by analogy with [6].

□