A Boundary Control Problem for a Nonlinear Parabolic Equation

Differential Equations, Vol. 39, No. 11, 2003, pp. 1626–1632. Translated from Differentsial’nye Uravneniya, Vol. 39, No. 11, 2003, pp. 1543–1549. c 2003 by Maksimov. Original Russian Text Copyright

PARTIAL DIFFERENTIAL EQUATIONS

A Boundary Control Problem for a Nonlinear Parabolic Equation V. I. Maksimov Institute for Mathematics and Mechanics, Urals Division, Russian Academy of Sciences, Yekaterinburg, Russia Received March 4, 2002

In the present paper, we continue the studies in [1, 2] and consider a robust control problem for a nonlinear parabolic system for the case in which the control and the perturbations occur both on the right-hand side of the equation and in the Dirichlet boundary conditions. The aim of the present paper is to describe a feedback boundary control method and justify it mathematically. Let us proceed to the statement of the problem. Consider the parabolic equation xt (t, η) − ∆L x(t, η) = f0 (t, η) + F (η, t, u1 , v1 ) + Φ(x(t, η))

in

T ×Ω = Q

(1)

with the initial condition x(0, η) = x0 (η)

in Ω

(2)

and the boundary condition x(t, σ) = ψ (σ, t, u2 , v2 )

in

T × ∂Ω.

(3)

Here T = [0, ϑ]; x0 (η) ∈ L2 (Ω); F (η, t, u1 , v1 ) = π(η)F (t, u1 , v1 ); ψ (σ, t, u2 , v2 ) = ω(σ)f (t, u2 , v2 ); Ω ⊂ Rq is a bounded domain with sufficiently smooth boundary ∂Ω; ∆L is the Laplace operator; f0 (·) ∈ L2 (T ; L2 (Ω)) is a given perturbation; Φ(·), f (· , · , ·), and F (· , · , ·) are Lipschitz functions; ω ∈ L2 (∂Ω); π(η) ∈ L2 (Ω). The system is subjected to controls u1 (·) and u2 (·) [u(t) = {u1 (t), u2 (t)} ∈ P = P1 × P2 for almost all t ∈ T , where P1 ⊂ U1 = RN1 and P2 ⊂ U2 = RN2 are bounded closed sets]. In addition, the system is subjected to uncontrolled perturbations v1 (·) and v2 (·) [v(t) = {v1 (t), v2 (t)} ∈ Z = Z1 × Z2 for almost all t ∈ T , where Z1 ⊂ V1 = RM1 and Z2 ⊂ V2 = RM2 are bounded closed sets]. The phase states x (τi , η) = x (τi ; 0, x0 , u(·), v(·)) ∈ H = L2 (Ω) of system (1)–(3) are measured at sufficiently frequent time instants τi ∈ T , τi = τi−1 + δ, i ∈ [1 : m − 1], τ0 = 0, τm = ϑ. The measurements ξi = ξ (τi ) ∈ H satisfy the inequalities |ξi − x (τi )|H ≤ h,

(4)

where h is the measurement accuracy parameter. The problem is to give a feedback control law u(t) = ue (t) = uei (ξi ) ∈ P , t ∈ [τi , τi+1 ), i ∈ [0 : m − 1], for system (1)–(3) such that for an arbitrary unknown perturbation v = v(t) ∈ Z, the phase state of the system x(t) = x (t; 0, x0 , u(·), v(·)) at time t = ϑ enters a sufficiently small ε-neighborhood of a given set M ⊂ H. A rigorous definition of a solution of Eq. (1) will be given below. The control law, that is, the law determining the evolution of the parameter u(t), is to be chosen by a “player.” (We use the terminology of the theory of positional differential games [3, 4].) The player should choose the control law so as to ensure the above-mentioned property of the motion for every possible realization of the perturbation v = v(t). We point out that the nature of v(t) is irrelevant to us. It can be a program control or a positional feedback control generated by somebody on the basis of some considerations. We only require that the following two conditions be satisfied: the realization v(t) must be a (Lebesgue) measurable function on the interval T and satisfy the inclusion v(t) ∈ Q for almost all t ∈ T . Throughout the following, by U = U1 × U2 we denote the control space, by V = V1 × V2 we denote the perturbation space, the symbol x (· ; 0, x0 , u(·), v(·)) stands for the solution of Eq. (1) with the initial condition (2), the boundary c 2003 MAIK “Nauka/Interperiodica” 0012-2661/03/3911-1626$25.00

A BOUNDARY CONTROL PROBLEM FOR A NONLINEAR PARABOLIC EQUATION

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condition (3), the control u(·), and the perturbation v(·), and M ε stands for the ε-neighborhood of M . Let us give some definitions. Following [1], we define an admissible control as an arbitrary (Lebesgue) measurable function u(·) : T 7→ P and an admissible perturbation as an arbitrary measurable function v(·) : T 7→ Q. The sets of all admissible controls and perturbations are denoted by U and V , respectively. A positional strategy is an arbitrary control law in which the control is chosen on the basis of measurements of the state x(t). We identify such a law with a pair (∆, U∆ ), m where ∆ = {τi }i=1 , τ0 = 0, τi = τi−1 + δ(∆), τm = ϑ, is a partition of the time interval T and U∆ : (τi , x) → Uτi ,τi+1 , τi ∈ ∆, x ∈ H, is some mapping (the feedback, or, in other words, the control choice rule). Here the symbol Uτi ,τi+1 stands for the restriction of the set U to the half-open interval [τi , τi+1 ). A motion of system (1) generated by a positional strategy (∆, U∆ ) together with a perturbation v(·) ∈ V and corresponding to a measurement accuracy parameter h > 0 is a continuous function xh (· ; U∆ , v(·)) defined by the rule
xh (t) = xh (t; U∆ , v(·)) = xh t; τi , xh (τi ) , ueτi ,τi+1 (·), vτi ,τi+1 (·) ,
xh (t) = p t; xh (τi ) , ueτi ,τi+1 (·), vτi ,τi+1 (·), xh (·) ∀t ∈ δi , t ∈ δi = [τi , τi+1 ) , ueτi ,τi+1 (s) ∈ U∆ (τi , ξi ) ,

ξi ∈ H,

|ξi − xh (τi )|H ≤ h.

The set of all such motions will be denoted by Xh (∆, U ). We say that a given motion xh (·) of system (1) solves the ε-guidance problem with target set M if xh (ϑ) ∈ M ε . Let σ be the Dirichlet mapping [5, 6], that is, the mapping given by the rule σu2 = z ⇐⇒

n

∆L z = 0 z = u2

in Ω in ∂Ω, u2 ∈ L2 (∂Ω),

or, equivalently, let z = σu2 be the solution of the variational equation Z Z z(η)∆L ψ(η)dη = (u2 (σ)∂ψ/∂n(σ)) dσ for all ψ ∈ D(A) = H01 (Ω) ∩ H2 (Ω). Ω

∂Ω

Here the symbol ∂ψ/∂n stands for the outward normal derivative, and H01 (Ω) and H2 (Ω) are the standard Sobolev spaces [7, pp. 28–30 of the Russian translation]. It is known [5, 6] that σ ∈ L (L2 (∂Ω); H). We introduce a mapping t → p(t; ·, ·, ·, ·) : H × L2 (T ; U × V ) × C(T ; H) → C(T ; H) by the formula Zt S(t − τ )σψ (· ; τ, u2 (τ ), v2 (τ )) dτ

p (t; x0 , u(·), v(·), z(·)) = S(t)x0 + A 0

Zt S(t − τ ) {f (τ ) + F (· ; τ, u1 (τ ), v1 (τ )) + Φ(z(τ ))} dτ,

+

t ∈ T.

0

Here the operator A given by Az = ∆z, z ∈ D(A), is the generator of a contraction semigroup of linear continuous operators {S(t); t ≥ 0} on H. By analogy with [1, 8], we define the solution of Eq. (1)–(3) corresponding to a control u(·) ∈ U and a perturbation v(·) ∈ V as the unique function x(·) = x (· ; 0, x0 , u(·), v(·)) ∈ C(T ; H) satisfying the integral equation z(t) = p (t; x0 , u(·), v(·), z(·)) ∀t ∈ T. It follows from [8] that, under our conditions, such a solution exists, is unique, and in addition has the semigroup property. DIFFERENTIAL EQUATIONS

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MAKSIMOV

We say that a family (∆h , U∆h ), h ∈ (0, 1), of positional strategies solves the robust control problem if, for each ε > 0, there exists an h∗ > 0 such that, for every h ∈ (0, h∗ ], any motion in the family Xh (∆h , U∆h ) solves the ε-guiding problem for the set M . Our aim is to construct such a family of strategies. The solution is essentially given in Theorem 1 below. We set \ Fu1 (t, v1 ) = co [F : F = F (t, u1 , v1 ) , u1 ∈ P1 ] , H(t) = Fu1 (t, v1 ) , v1 ∈Z1



H(·) = u1 (·) ∈ L2 T ; L2 Ω; RN1 : u1 (η, t) = π(η)˜ u1 (t), u ˜1 (t) ∈ H(t) for almost all t ∈ T , \ ψu2 (t, v2 ) = co [ψ : ψ = ψ (t, u2 , v2 ) , u2 ∈ P2 ] , R(t) = ψu2 (t, v2 ) , v2 ∈Z2



R(·) = u2 (·) ∈ L2 T ; L2 ∂Ω; RN2 : u2 (σ, t) = ω(σ)˜ u2 (t), u˜2 (t) ∈ R(t) for almost all t ∈ T . Here co D stands for the closed convex hull of a set D. We introduce the following condition. Condition. (a) The sets H(t) and R(t) are nonempty for any t ∈ T . (b) There exists a control u∗ (·) = {u∗1 (·), u∗2 (·)} ∈ H(·) × R(·) that brings the phase trajectory of the system
x0t (t, η) − ∆L x0 (t, η) = f (t, η) + u∗1 (t, η) + Φ x0 (t, η) in Q (5) with the initial condition and the boundary condition

x0 (0, η) = x0 (η) x0 (t, σ) = u∗2 (t, σ)

in

Ω

in T × ∂Ω

to the set M at time ϑ. (c) The small-game saddle point conditions are satisfied (s = ±1): min max sf (t, u2 , v2 ) = max min sf (t, u2 , v2 ) ,

t ∈ T,

(6)

min max sF (t, u1 , v1 ) = max max sF (t, u1 , v1 ) ,

t ∈ T.

(7)

u2 ∈P2 v2 ∈Z2

v2 ∈Z2 u2 ∈P2

u1 ∈P1 v1 ∈Z1

v1 ∈Z1 u1 ∈P1

The auxiliary problem (5) (more precisely, the “stable path” in the terminology of [3, p. 207]) is an analog of the well-known guide in the theory of positional differential games. It is essentially a mental construction helping the player to form the desired control of the actual system in the course of operation. m Let us describe the solution algorithm. Let a family {∆h } of partitions ∆h = {τi }i=0 , τi = τh,i , m = mh , τ0 = 0, tm = ϑ, of the interval T with diameters δ = δ(h) be chosen. Before the algorithm m starts, we choose a number h ∈ (0, 1) and the partition ∆h = {τi }i=0 , τi = τhi , m = mh , of diameter δ = δ(h) = δ (∆h ). Then we organize a feedback control of the real system (1) such that, for sufficiently small h and δ, the motion xh (·) = xh (· ; U∆h , v(·)) of system (1) enters a sufficiently small neighborhood M ε of the set M at time ϑ for an arbitrary implementation v(·) ∈ V . To this end, we fix h and ∆h before the algorithm starts. The algorithm splits into m − 1 similar steps. At the ith step, the following operations are performed on the time interval δi = [τi , τi+1 ). We first compute the controls ue1i and ue2i according to the rules   ∂ −1 ∗ max f (τi , ue2i , v2 ) A si , ω v2 ∈Z2 ∂n ∂Ω L2 (∂Ω) ( ) (8)   ∂ −1 ∗ ≤ inf max f (τi , u2 , v2 ) + hδ, A si , ω u2 ∈P2 v2 ∈Z2 ∂n ∂Ω L2 (∂Ω)
max s∗i , A−1 π H F (τi , ue1i , v1 ) v1 ∈Z1   (9)
∗ −1 ≤ inf max si , A π H F (τi , u1 , v1 ) + hδ, u1 ∈P1

v1 ∈Z1

DIFFERENTIAL EQUATIONS

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where s∗i = A−1 (ξi − ψi ) and |ψi − x0 (τi )|H ≤ h. After that, the constant controls u1 (t) = ue1i and u2 (t) = ue2i , t ∈ δi , are fed into system (1) on the time interval δi ; i.e., we set

U∆

h

(τi , ξi ) = {ue1i , ue2i } ,

t ∈ δi .

(10)

These two controls and some unknown perturbations v1 (t) and v2 (t), t ∈ δi , bring system (1) from the state xh (τi ) to the state xh (τi+1 ). At the (i + 1)st step, similar actions are performed. The algorithm terminates at time t = ϑ. Theorem. The family (∆h , U∆h ) , h ∈ (0, 1), of positional strategies (8)–(10) solves the robust control problem. Before proving the theorem, we present some auxiliary assertions. Lemma 1. The pencil XT = {x (· ; 0, x0 , u(·), v(·)) : u(·) ∈ U , v(·) ∈ V } of solutions of (1)–(3) is bounded in the metric of the space C(T ; H). The assertion of the lemma follows from the boundedness of the sets U and V , the Lipschitz property of the function Φ, inequality (3.14) in [6] [ |AS(t)σω∗ |H ≤ ct−7/8 |ω∗ |L2 (∂Ω) , t > 0, ω∗ ∈ L2 (∂Ω)], and the fact that {S(t); t ≥ 0} is a contraction semigroup. 2 We set εh (t) = |A−1 (xh (t) − x0 (t))|H and ϕ(δ) = max {ωj (δ) : j ∈ [1 : 3]}, where ω1 (δ) = sup |S(δ − τ )σω − σω|H → 0, τ ∈[0,δ]

ω2 (δ) = ω3 (δ) =

sup

sup |f (t, u2 , v2 ) − f (t + τ, u2 , v2 )| → 0,

sup

sup |F (t, u1 , v1 ) − F (t + τ, u1 , v1 )| → 0 as

τ ∈[0,δ] u2 ∈P2 t∈[0,ϑ−δ] v2 ∈Z2 τ ∈[0,δ] u1 ∈P1 t∈[0,ϑ−δ] v1 ∈Z1

δ →0+.

For x(·) ∈ XT , by Ξh (x(·)) we denote the set of all piecewise constant functions ξ(t) : T → H such that supt∈T |ξ(t) − x(t)|H ≤ h. The set Ξh (x0 (·)) for the solution x0 (·) of Eq. (5) is defined in a similar way. Lemma 2. The inequality εh (t) ≤ k(δ + h + ϕ(δ)),

t ∈ T,

(11)

is valid uniformly with respect to all ξ(·) ∈ Ξh (xh (·)) , ψ(·) ∈ Ξh (x0 (·)) , h ∈ (0, 1), and partitions m ∆ = {τi }i=0 of the interval T with diameters δ ≤ 1. Proof. Let us estimate the variation of the quantity εh (t) on the interval δi = [τi , τi+1 ]. By v1 (t) and v2 (t) we denote the (unknown) perturbations acting on system (1) on the interval δi . We find e e the vectors v1i and v2i from the conditions   ∂ −1 ∗ e min f (τi , u2 , v2i ) A si , ω u2 ∈P2 ∂n ∂Ω L2 (∂Ω) ( )   ∂ −1 ∗ ≤ sup min f (τi , u2 , v2 ) + hδ, A si , ω u2 ∈P2 ∂n v2 ∈Z2 ∂Ω L2 (∂Ω)  

∗ −1 e ∗ −1 min si , A π H F (τi , u1 , v1i ) ≤ sup min si , A π H F (τi , u1 , v1 ) + hδ. u1 ∈P1

v1 ∈Z1

u1 ∈P1

By the Carath´eodory theorem [3, p. 195] and Condition (a), we have the representations ! N N 1 +1 1 +1   X X (1) (r) (1) (1) (r) e u∗1 (t, η) = π(η), αt,r ≥ 0, αt,r F t, u1,t , v1i αt,r = 1, u1,t ∈ P1 , r=1

u∗2 (t, σ) =

N 2 +1 X

!   (2) (p) e ω(σ), αt,p f t, u2,t , v2i

r=1 (2)

αt,p ≥ 0,

p=1

DIFFERENTIAL EQUATIONS

N 2 +1 X p=1

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(2)

αt,p = 1,

(p)

u2,t ∈ P2 .

1630

MAKSIMOV

One can readily see that  t−τ Z i −1 
0 ε(t) = A S (t − τi ) xh (τi ) − x (τi ) + A S (t − τi − τ ) %i (τ ; ·)dτ  0

 2 
  0 S (t − τi − τ ) µi (τ ; ·) + Φ (xh (τi + τ ; ·)) − Φ x (τi + τ ; ·) dτ 

t−τ Z i

+ 0

≤

3 X

Jjit

(12)

H

for

t ∈ [τi , τi+1 ) ,

j=1

where 2 t J1i = sti H , 


sti = A−1 S (t − τi ) xh (τi ) − x0 (τi ) ,  t−τ Z i t J2i = 2 sti , S (t − τi − τ ) %i (τ ; ·)dτ  , 

0

t J3i = 2 sti , A−1

H

t−τ Z i

S (t − τi − τ ) µi (τ ; ·)dτ  ,

0

 t J4i = 2 sti , A−1





H

t−τ Z i








S (t − τi − τ ) Φ x0 (τi + τ ; ·) − Φ (x (τi + τ ; ·)) dτ  ,

0

H

 t−τ 2 2 t−τ Z i  Z i t −1 J5i = 3 S (t − τi − τ ) %i (τ, ·)dτ + A S (t − τi − τ ) µi (τ ; ·)dτ  0 0 H H  2 t−τ Z i −1 
 0 , + A S (t − τi − τ ) Φ x (τi + τ ; ·) − Φ (x (τi + τ ; ·)) dτ  0 !H N 1 +1   X (1) (r) e µi (τ ; ·) = F (τi + τ, ue1i , v1 (τi + τ )) − π(·), ατi +τ,r F τi + τ, u1,τi +τ , v1i r=1

%i (τ ; ·) =

f (τi +

τ, ue2i , v2

(τi + τ )) −

N 2 +1 X

(2) ατi +τ,p f

!   (p) e (σω)(·). τi + τ, u2,τi +τ , v2i

p=1

Since {S(t); t ≥ 0} is a contraction semigroup, the operator A−1 commutes with S(δ), and Φ(·) is a Lipschitz function, we have the inequalities t J1i ≤ ε (τi ) ,

(13)


t J4i ≤ 2c A−1 xh (τi ) − x0 (τi ) H

t−τ Z i

−1
A xh (τi + τ ) − x0 (τi + τ ) H dτ

0

(14)

Zt 2

≤ (t − τi ) c2 ε (τi ) + (t − τi ) c = L A−1

L (H;H)

ε(τ )dτ ≤ (t − τi ) c2 ε (τi ) + k0 (t − τi ) , τi

|A|L (H;H) . DIFFERENTIAL EQUATIONS

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A BOUNDARY CONTROL PROBLEM FOR A NONLINEAR PARABOLIC EQUATION

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Here L is the Lipschitz constant of Φ(·). Furthermore, since the sets P1 and P2 are bounded, A−1 ∈ L (H; H), and σ ∈ L (L2 (∂Ω); H), it follows from Lemma 1 that t J5i ≤ k1 δ2 ,

(15)

where k1 = const ∈ [0, +∞). Note that [8] Zt S(t − s)x ds = S(t)x − x

A

∀x ∈ H.

0

Therefore,

−1 A {S(t)x − x} ≤ t|x|H . H

(16)

Further, with regard to the inclusions ξ(·) ∈ Ξh (xh (·)) and ψ(·) ∈ Ξh (x (·)) and inequality (16), we obtain t 
si − s∗i = A−1 S (t − τi ) xh (τi ) − x0 (τi ) − (ξi − ψi ) ≤ k2 (h + t − τi ) . (17) H H 0

Consequently, t−τ Z i t J2i

(s∗i , S (t − τi − τ ) %i (τ ; ·))H dτ + k3 (t − τi ) {h + t − τi }

≤2

0 t−τ Z i

≤2

s∗i , f (τi , ue2i , v2 (τi + τ )) −

N 2 +1 X

(2) ατi +τ,p f



(p) e τi , u2,τi +τ , v2i

p=1

0

!

 σω

dτ H

+ k4 (t − τi ) (h + t − τi + ω1 (t − τi ) + ω2 (t − τi )) . According to [5],

∂ −1 σ x= A x ∂n ∂Ω ∗

∀x ∈ H.

(18)

e Note that, by the definition of the vectors ue2i [by (8)] and v2i and by the saddle point condition (6), we have the inequalities   ∂ −1 ∗ f (τi , ue2i , v2 (τi + τ )) A si , ω ∂n ∂Ω L2 (∂Ω)     ∂ −1 ∗ (p) e ≤ , τ ∈ δi . f τi , u2,τi +τ , v2i A si , ω ∂n ∂Ω L2 (∂Ω) (2)

By multiplying these inequalities by ατi +τ,p and by summing the resulting relation with respect to p, we obtain   ∂ −1 ∗ f (τi , ue2i , v2 (τi + τ )) A si , ω ∂n ∂Ω L2 (∂Ω) (19)   N 2 +1   X ∂ −1 ∗ (2) (p) e ≤ ατi +τ,p f τi , u2,τi +τ , v2i , τ ∈ δi . A si , ω ∂n ∂Ω L2 (∂Ω) p=1 Therefore, relations (18) and (19) imply that Zδ  t J2i ≤2 0

 ∂ −1 ∗ A si , ω ∂n ∂Ω L2 (∂Ω)

f (τi , ue2i , v2 (τi + τ )) −

N 2 +1 X

(2) ατi +τ,p , f



(p) e τi , u2,τi +τ , v2i

!  dτ

p=1

+ k4 (t − τi ) (h + t − τi + ω1 (t − τi ) + ω2 (t − τi )) ≤ k4 (t − τi ) (h + t − τi + ω1 (t − τi ) + ω2 (t − τi )) . (20) DIFFERENTIAL EQUATIONS

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MAKSIMOV

Likewise, by using (7), (16), and (17), we obtain     t−τ t−τ Z i Z i 2 t J3i = 2 si , A−1 S (t − τi − τ ) µi (τ ; ·)dτ  ≤ 2 si , A−1 µi (τ ; ·)dτ  + k4 (t − τi ) 

0

≤ 2 s∗i , A−1 π

0

H

t−τ Z i

F (τi , ue1i , v1 (τi + τ )) −

N 1 +1 X

!    (r) e dτ  α(1) τi +τ,r F τi , u1,τi +τ , v1i H

r=1

0

H

+ k5 (t − τi ) (t − τi + h + ω3 (t − τi )) . Therefore, by (9) and (10), we have the inequality t J3i ≤ k5 (t − τi ) (t − τi + h + ω3 (t − τi )) .

(21)

By combining the estimates (12)–(15), (20), and (21), we obtain
ε(t) ≤ 1 + (t − τi ) c2 ε (τi ) + k6 (t − τi ) (t − τi + h + ϕ (t − τi )) ,

t ∈ [τi , τi+1 ] .

This, together with the standard technique in [3, pp. 59–61], implies that 2 ε(t) ≤ A−1 (x0 − w0 ) H + k7 (δ + h + ϕ(δ)) ≤ k(δ + h + ϕ(δ)),

t ∈ T.

The proof of the lemma is complete. Remark 1. If system (1) is linear in the phase variable x, i.e., Φ ≡ 0, then the conditions imposed on the measurement results ξi can be weakened. More precisely, inequalities (4) can be replaced by the inequalities |ξi − x (τi )|H −1 (Ω) ≤ h. In this case, the theorem remains valid. Remark 2. We have considered only the encounter problem. One can readily see that, by using the scheme in the monograph [4], one can point out an algorithm for the evasion problem and prove the theorem on the alternative. ACKNOWLEDGMENTS The work was financially supported by the Russian Foundation for Basic Research (grant no. 01-01-00566). REFERENCES 1. Osipov, Yu.S., Pandolfi, L., and Maksimov, V.I., Sbornik dokl. mezhdunar. konf. “Raspredelennye sistemy: optimizatsiya i prilozheniya v ekonomike i naukakh ob okruzhayushchei srede” (DSO’2000), Yekaterinburg, 30 maya–2 iyunya 2000 g. (Proc. Int. Cong. “Distributed Systems: Optimization and Applications in Economics and Environmental Sciences, DSO0 2000. Yekaterinburg, Russia, May 30–June 2, 2000), Yekaterinburg, 2002. 2. Osipov, Yu.S., Pandolfi, L., and Maksimov, V.I., Dokl. RAN , 2000, vol. 374, no. 3, pp. 310–312. 3. Krasovskii, N.N., Upravlenie dinamicheskoi sistemoi (Control of a Dynamical System), Moscow, 1985. 4. Krasovskii, N.N. and Subbotin, A.I., Pozitsionnye differentsial’nye igry (Positional Differential Games), Moscow, 1974. 5. Lasiecka, I., Lions, J.-L., and Triggiani, R., J. Math. Pures et Appl., 1986, vol. 65, no. 2, pp. 227–243. 6. Barbu, V., SIAM. J. Control and Optimiz., 1980, vol. 18, no. 2, pp. 227–243. 7. Lions, J.-L., Contrˆ ole optimal de syst´emes gouvern´es par des ´equations aux d´eriv´ees partielles, Paris: Dunod, 1968. Translated under the title Optimal’noe upravlenie sistemami, opisyvaemymi uravneniyami s chastnymi proizvodnymi, Moscow: Mir, 1972. 8. Lasiecka, I. and Triggiani, R., Appl. Math. and Optimiz., 1991, vol. 23, no. 2, pp. 109–154. 9. Ioffe, A.D. and Tikhomirov, V.M., Teoriya ekstremal’nykh zadach (Theory of Extremal Problems), Moscow, 1974.

DIFFERENTIAL EQUATIONS

Vol. 39

No. 11

2003