ISSN 0965-5425, Computational Mathematics and Mathematical Physics, 2016, Vol. 56, No. 1, pp. 14–25. © Pleiades Publishing, Ltd., 2016. Original Russian Text © V.I. Maksimov, Yu.S. Osipov, 2016, published in Zhurnal Vychislitel’noi Matematiki i Matematicheskoi Fiziki, 2016, Vol. 56, No. 1, pp. 16–28.
Infinite-Horizon Boundary Control of Distributed Systems V. I. Maksimova and Yu. S. Osipovb a Krasovskii
Institute of Mathematics and Mechanics, Ural Branch, Russian Academy of Sciences, ul. S. Kovalevskoi 16, Yekaterinburg, 620219 Russia Ural Federal University, ul. Mira 19, Yekaterinburg, 620002 Russia b Presidium of the Russian Academy of Sciences, Leninskii pr. 32a, Moscow, 119991 Russia Faculty of Computational Mathematics and Cybernetics, Moscow State University, Moscow, 119992 Russia e-mail:
[email protected] Received June 6, 2015
Abstract―For a boundary controlled dynamic system, algorithms for solving the problem of tracking reference motion and the problem of tracking reference control are described. The algorithms are robust to information noise and computational errors. The solution method is based on the extremal shift method from the theory of positional differential games. Keywords: distributed control systems on infinite horizon, boundary control, computational algorithm, estimation of computational errors, extremal shift method. DOI: 10.1134/S0965542516010139
1. INTRODUCTION AND FORMULATION OF THE PROBLEMS For a boundary controlled system, we consider two problems: tracking reference motion and tracking a reference control. The solution method is based on the extremal shift method, which is well known in the theory of positional differential games. Let us describe the problems in more detail. In a Hilbert space (H, |·|H), we consider a dynamic system with a phase trajectory described by the equation t
∫
x(t;0, u(⋅)) = S (t )x 0 − A S (t − s)Du(s)ds,
t ∈ T = [0, +∞).
(1.1)
0
Here, A is an infinitesimal generator of the C0-semigroup of linear continuous operators {S(t): t ≥ 0} in H and (U, |·|U) is a uniformly convex separable Banach space. The properties of the operator D will be specified later. Assume that {S(t): t ≥ 0} is a contraction semigroup, i.e.,
S (t ) +(H ;H ) ≤ 1 and let {S(t): t ≥ 0} be exponentially stable: |S(t)|+(H; H) → 0
∀ t ≥ 0,
exponentially as
t → +∞.
This condition implies that the operator A–1 is bounded, which will be used later. Here and below, |S|L(H; H)
denotes the norm of a linear continuous operator from H to H: S ∈ L(H; H). Note that the integral on the right-hand side of (1.1) generally does not belong to the domain of A. Therefore, it is natural to indicate the classes of systems for which formula (1.1) does characterize the phase trajectory of an actual system. First, formula (1.1) makes sense if D = –A–1B, B ∈ L(U; H). In this case, Eq. (1.1) describes the output of the system x = Ax + Bu with a distributed input action u(·). Second, in [1] two classes of objects corresponding to parabolic and hyperbolic systems with a boundary input were chosen for which formula (1.1) is also valid. 14
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The first is the class of systems with holomorphic semigroups and a weakly regular linear operator D (i.e., an operator whose range R(D) belongs to the domain of –A with a fractional exponent):
R(D) ⊆ D((− A) γ ),
0 < γ < 1.
A typical example of systems of this class is the heat equation with a Dirichlet boundary condition:
x ' = Δ L x on Ω,
x(t, ⋅) = u(t, ⋅) on Γ.
Here, Ω is a bounded open domain in Rn with a smooth boundary Γ. This equation can be formalized in the form of (1.1) (see [2, 3]) if Ω is sufficiently regular. It should be kept in mind that, in the case of an unbounded control, i.e., for u(·) ∈ L2(T; U), the function x(·) defined by formula (1.1) is generally only square integrable. Therefore, point observation (at times τi ∈ T) is not possible. However, if we assume that the admissible inputs are bounded (u(t) ∈ P), then the trajectory t → x(t; 0, x0, u(·))is continuous (see [3, Remark 2]). The second class (see [1]) consists of dynamic systems satisfying the trace condition θ
∫ D*A*S
2 A * (t ) x H
dt ≤ const ⋅ x H
∀ x ∈ D( A1* ),
2
∀ϑ > 0,
0
where A* denotes the adjoint operator and {S A*(t): t ≥ 0} is the semigroup of linear operators generated by A*. If this condition holds, then the function t → x(t; 0, x0, u(·)) is continuous for any square integrable input u(·) (see [4]). A typical representative of this class is the wave equation (with damping, if we want exponential stability). Note the following important fact: u(·) → x(·; 0, x0, u(·)) is a continuous mapping from the space L2([0, ϑ]; U) (ϑ ∈ (0, +∞)) to L2([0, ϑ]; H) (in the first (“parabolic”) case) and to C([0, ϑ]; H) (in the second (“hyperbolic”) case). Formally, the differential equation whose motion is given by formula (1.1) can be written as
x = A( x − Du). It should be stressed that this equation is convenient for describing systems with boundary or distributed inputs. If −1
D = D ' − A D '', then the operator D" corresponds to a distributed input, while D', to a boundary input. The problems under study are considered from the point of view of a controller, which generates controls. Let us describe the formulations of these problems. In what follows, D: U → H is assumed to be a linear bounded operator such that formula (1.1) makes sense for any function u(·) ∈ L∞(T; U); moreover, the semigroup property holds for such u(·). Specifically, the function t → x(t; 0, x0, u(·)) ∈ H is continuous and
x(t; τ, x(τ), u(⋅)) = x(t;0, x 0, u(⋅))
∀ 0 < τ < +∞,
∀ t ∈ [τ, +∞).
Here, x(τ) = x(τ; 0, x0, u(·)) is the state of system (1.1) at the time τ generated by the control u(·) ∈ L∞([0, τ], U), u(t) ∈ P, where P ⊂ U is a bounded and closed set. On the time interval T, we define a family Δ = (τ i )i∞=0 of observation times τi + 1 = τi + τi, i = 0, 1, …, where τ0 = 0 and δi > 0. Assume that, at each time τi, i = 0, 1, …, we observe the state x(τi) = x(τi, 0, x0, u(·)) ∈ H of system (1.1) and the observation result is an element ξ ih ∈ H having the property
| ξ ih − x(τ i )| H ≤ ν ih,
(1.2)
where h ∈ (0, 1) is an upper threshold of the measurement error and ν ih ∈ (0, h) is the measurement error at the time τi. COMPUTATIONAL MATHEMATICS AND MATHEMATICAL PHYSICS
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1.1. Problem of Tracking Reference Motion Consider a reference motion w(·) described by the relation t
∫
w(t;0, w0, u(⋅)) = S (t )w0 − A S (t − s)Du(s)ds,
t ∈T,
(1.3)
0
with u(·) = u*(·), where u*(·) is a reference control. Here, u*(·) and the corresponding reference motion w(·) are previously unknown. A priori information on the reference control is that it takes values in a given bounded closed set P ⊂ U. At each time τi, i = 1, …, the controller receives the current reference state w(τi; 0, w0, u*(·)). For system (1.1), the goal is to design an algorithm that, relying on the information received at the observation times, generates current control values u(t) = ui(ψ ih , ξ ih) ∈ P, t ∈ [τi, τi + 1), such that the corresponding motion x(·) of system (1.1) belongs to a “small” neighborhood of the reference motion w(·). The states w(τi) of the reference system can also be measured with an error. In this case, u(t) = ui(ψ ih , ξ ih) ∈ P for t ∈ [τi, τi + 1), where ψ ih ∈ H is a measurement of w(τi) such that
| ψ ih − w(τ i )| H ≤ ν ih.
(1.4)
1.2. The Problem of Tracking a Reference Control Under the conditions of the preceding problem, for system (1.1), the task is to construct an algorithm that, relying on the information received at the observation times, generates control values u(t) = ui( ψ ih , ξ ih) ∈ P for t ∈ [τi, τi + 1) such that the resulting control implementation u(·) belongs to a “small” neighborhood of the reference control u*(·). Let us give formal definitions. For every h ∈ (0, 1), let Δh be a fixed family of partitions of the interval T by control times τh, i: ∞
Δ h = {τ h,i }i =0,
+∞
τ h,0 = 0,
τ h,i +1 = τ h,i + δ i ,
δ i = δ i (h) ∈ (0,1),
δ i (h) ∈ (0, h),
∑ δ (h) = +∞. (1.5) i
i =0
Any piecewise constant function
ξ h(⋅) : T H ,
ξ h(t ) = ξ ih
t ∈ [τ h,i , τ h,i +1),
fo r
i = 0,1,… ,
satisfying inequalities (1.2) is called an admissible measurement of x(·) of accuracy h, functions u(·) ∈ L∞(T; U) are called admissible controls, and functions
8 α (,⋅ ⋅, ⋅) : T × H × H U ,
9(,⋅ ⋅, ⋅) : T × H × H U
(1.6)
are called admissible feedbacks (for system (1.1)). Here, α > 0 is a parameter; its interpretation will be given later. Similarly, any piecewise constant function
ψ h(⋅) : T H ,
ψ h(t ) = ψ ih
t ∈ [τ h,i , τ h,i +1),
fo r
i = 0,1,… ,
satisfying inequalities (1.4) is called an admissible measurement of w(·) of accuracy h. Consider the second problem first. The control u = uα, h(·) is specified using the feedback 8α(·, ·, ·). The phase trajectory x = xα, h(·) of system (1.1) is then observed at the discrete times τh, i with an error and varies under the action of some feedback uα, h(·) = 8α(·, ξh(·), ψh(·)). Thus, this trajectory depends on the measurements ξh(·) of xα, h(·) (i.e., on admissible measurements of xα, h(·) of accuracy h) and on the measurements ψh(·) of w(·) (admissible measurements of w(·) of accuracy h) and is determined by the relation t
x α,h(t ) = x(t;0, x 0, u α,h(⋅)) = S (t )x 0 − A S (t − s)Du α,h(s)ds,
∫
t ∈T,
(1.7)
0
where
u
α,h
α,h
(t ) = ui (t ) = 8 α (τ i , ξ i , ψ i )
τ i = τ i , h,
i = 0,1,… ,
h
h ξi
h
for
= ξ (τ i ), h
t ∈ [τ i , τ i +1) ,
(1.8)
ψ i = ψ (τ i ) . h
h
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For any admissible feedback 8α(·, ·, ·) and any admissible measurements ξh(·) and ψh(·) of accuracy h, the function xα, h(·) given by (1.7) is called the h-trajectory of actual system (1.1) corresponding to the admissible feedback 8α(·, ·, ·) and the admissible measurements ξh(·) and ψh(·) of accuracy h. The controllable xα, h-process corresponding to the admissible feedback 8α(·, ·, ·) and admissible measurements of accuracy h is any quintuple (w(·), ξh(·), ψh(·), xα, h(·), uα, h(·)), where w(·) is reference motion (1.3), ξh(·) is an admissible measurement of xα, h(·) of accuracy h, ψh(·) is an admissible measurement of w(·) of accuracy h, xα, h(·) is the trajectory of system (1.1) corresponding to 8α(·, ·, ·), ξh(·), and ψh(·) (i.e., function (1.7)), and the control u = uα, h(·) : T ° U is specified according to (1.8). The function uα,h(·) is called the h-implementation of the admissible feedback 8α(·, ·, ·) corresponding to admissible measurements of accuracy h. The basic element of the solution to the problem under study is the admissible feedback 8α(·, ·, ·). This feedback is called reconstructing if there is a number h0 ∈ (0, 1) and functions γU(·) : (0, 1) → [0, +∞), γ1(·) : (0, 1) → [0, +∞), and α = α(h) : (0, 1) → (0, 1) such that γU(h) → 0, γ1(h) → 0, α(h) → 0 as h → 0. Moreover, for any h ∈ (0, h0), any family Δh of partitions of T, any h-implementation u = uα, h(·) of 8α(·, ·, ·) of form (1.6), any h-trajectory xα, h(·) of actual system (1.1) (i.e., function (1.7)) corresponding to a control u = uα, h(·) of form (1.8), and any admissible measurements ξh(·) and ψh(·) of accuracy h, we have −1
A (x
α,h
(τ h,i ) − w(τ h,i ))
τ h,i
∫
H
≤ C1γU (h)(1 + τ h,i ),
(1.9)
τ h,i 2
u α,h(s) ds ≤ U
0
∫ u*(s)
2 U
ds + C 2 γ 1(h).
(1.10)
0
Thus, inequalities (1.9) and (1.10) hold for the controllable h-process (w(·), xα, h(·), ξh(·), ψh(·), uα, h(·)), where C1 and C2 are constants. The functions γU(·) and γ1(·) are called error estimates of the admissible feedback 8α(·, ·, ·). Let us explain why γU(·) and γ1(·) were used as criteria for the deviation of uα, h(·) from u*(·). For a fixed ϑ > 0, let P be a convex set and Ωϑ be the set of all measurable functions u = u(·) with u(t) ∈ P for a.e. t ∈ [0, ϑ] that generate a solution x(·) of system (1.1). Suppose that u*(·) is a unique L2([0, ϑ]; U)-norm minimum element of Ωϑ (if any). Then it easy to prove that
u α,h(⋅) → u*(⋅)
in
L2 ([0, ϑ];U )
as
h → 0.
Consider the first problem. The control u = u h (·) is specified using the feedback 9(·, ·, ·) (see (1.6)). In this case, the phase trajectory x = xh(·) = x(t; 0, x0, u h (·)) of system (1.1) is also observed at the discrete times τh, i with an error. Moreover, it is affected by the feedback u h (·) = 9(·, ξh(·), ψh(·)). Thus, this trajectory depends on the results ξh(·) of measuring xh(·) (i.e., on admissible measurements of xh(·) of accuracy h) and on the results ψh(·) of measuring w(·) (admissible measurements of w(·) of accuracy h) and is determined by the relation t
∫
x (t ) = x(t;0, x 0, u (⋅)) = S (t )x 0 − A S (t − s)Du h(s)ds, h
h
t ∈T,
(1.11)
0
where
u h(t ) = uih(t ) = 9(τ i , ξ ih, ψ ih )
for
t ∈ [τ i , τ i +1),
τ i = τ i,h,
i = 0,1,….
(1.12)
For any admissible feedback 9(·, ·, ·) and any admissible measurements ξh(·) and ψh(·) of accuracy h, the function xh(·) = x(t; 0, x0, u h (·)) given by (1.11) is called the h-trajectory of the actual system corresponding to the admissible feedback 9(·, ·, ·) and to the admissible measurements ξh(·) and ψh(·) of accuracy h. The controllable xh-process corresponding to the admissible feedback 9(·, ·, ·) and admissible measurements of accuracy h is any quintuple (w(·), ξh(·), ψh(·), xh(·), u h (·)), where w(·) is reference motion (1.3), ξh(·) is an admissible measurement of xh(·) of accuracy h, ψh(·) is an admissible measurement of w(·) of accuracy h, xh(·) is the trajectory of system (1.1) corresponding to 9(·, ·, ·), ξh(·), and ψh(·) (i.e., funcCOMPUTATIONAL MATHEMATICS AND MATHEMATICAL PHYSICS
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tion (1.11)), and the control u h (·): T ° U is given by relation (1.12), where ξ ih = ξh(τi), ψ ih = ψh(τi) for t ∈ [τi, τi + 1), τi = τh, i, i = 0, 1, …. The function u h (·) is called the implementation of the admissible feedback 9(·, ·, ·) corresponding to admissible measurements of accuracy h. A tool for solving the tracking problem for reference motion is the admissible feedback 9(·, ·, ·). This feedback is called tracking if there exists a number h1 ∈ (0, 1) and a function γV(·) : (0, 1) ° [0, +∞) such that γV(h) → 0 as h → 0 and, given any h ∈ (0, h1), any family Δh of partitions of T, any implementation u = u h (·) of the admissible feedback 9(·, ·, ·) (see (1.6)), any trajectory xh(·) of system (1.1) (i.e., function (1.11)) corresponding to a control u h (·) of form (1.12), and any admissible measurements ξh(·) and ψh(·) of accuracy h, we have
| A −1( x h(τ h,i ) − w(τ h,i ))| H ≤ γ V (h),
i = 0,1,… ,
i.e., these inequalities hold for the controllable process (w(·), ξh(·), ψh(·), xh(·), u h (·)). The function γV(·) is called an error estimate of the admissible feedback 9(·, ·, ·). The tracking problems for reference motion and reference control are to construct a tracking and a reconstructing admissible feedback 9 and 8α, respectively. If the interval T is bounded, the problems can be solved using the constructions from [5–7]. It should be noted that the algorithms proposed in [5–7] are intended for a finite time interval. As the interval length increases, the computational and measurement errors accumulate. Algorithms free of this shortcoming were constructed in [8, 9] as applied to a system governed by a nonlinear (in the state variable) vector ordinary differential equation or a parabolic equation with a distributed input. In this paper, we present algorithms for solving the above problems for a distributed system with a boundary control. In recent years, much interest has been shown in the infinite-horizon control of distributed systems. Moreover, many of publications on this subject were devoted to feedback control. We note only some of them, where a relevant bibliography can be found. The problem of constructing a boundary control minimizing a quadratic cost function was considered in [10]. Its solution was found using the apparatus of Sobolev spaces in conjunction with the theory of Riccati equations. The feedback control of a system of distributed FitzHugh–Nagumo-type equations was discussed in [11]. Stabilizing controls were constructed that ensure the convergence of the solution of an evolutionary system to that of a stationary system. A Timoshenko-type system was considered in [12]. A feedback boundary control ensuring the decay of the energy functional was constructed by applying the method of multipliers. Additionally, the rate of decay was indicated. 2. ALGORITHM FOR SOLVING THE PROBLEM OF TRACKING REFERENCE MOTION Before describing a solution algorithm for the problem of tracking reference motion, we introduce two conditions to be used in what follows. Condition 1. The family Δh is such that +∞
∑ δ (h) ≤ ϕ (h), 2 i
ϕ1(h) → 0,
1
h → 0.
i =0
Condition 2. It holds that +∞
(ν 0h ) 2 +
∑ ν δ (h) ≤ ϕ (h), h i i
2
ϕ 2(h) → 0,
h → 0.
i =0
In what follows, we assume that the operators {S(t): t ≥ 0} are self-adjoint. Let u*(·) ∈ L∞(T; U) be a reference control taking values in a given bounded closed set P ⊂ U and w(·) be the reference motion of system (1.3) that corresponds to u*(·). Before starting the algorithm, we fix h ∈ (0, 1), a sequence {ν ih}i∞=0 , and a partition Δh = {τ h,i }i∞=0 (see (1.5)). The algorithm consists of same-type steps. The following operations are executed at the ith step, which is performed on the time interval [τi, τi + 1), τi = τh, i. First, at the time τi, i = 0, 1, …, when COMPUTATIONAL MATHEMATICS AND MATHEMATICAL PHYSICS
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the signal ξ ih ∈ H on the state of system (1.1) and the signal ψ ih ∈ H on the reference state of system (1.3) are received, we choose the function u h (t) determined by condition (1.12), where τ i +1 ⎧ ⎪ h = ⎨v τi ,τi +1 (⋅) ∈ Pτi ,τi +1 (⋅) : (qih(t − τ i ), Dv h(t )) H dt ⎪⎩ τi τ i +1 ⎧ ⎫ ⎫ ⎪ ⎪ h h ⎪ ≤ inf ⎨ (qi (t − τ i ), Dv(t )) H dt : v τi ,τi +1 (⋅) ∈ Pτi ,τi +1 (⋅)⎬ + d ν i δ i ⎬ . ⎪⎩ τi ⎪⎭ ⎪⎭
∫
9(τ i , ξ ih, ψ ih )
(2.1)
∫
Here, v τhi ,τi +1 (⋅) denotes the function vh(t), t ∈ [τi, τi + 1), d = const > 0, (·, ·)H is the inner product in H,
Pτi ,τi +1 (⋅) = {v τi ,τi +1 (⋅) : v(t ) ∈ P qih(t )
−1
t ∈ [τ i , τ i +1)},
for a.e.
= A S (2δ i −
t )(ψ ih
−
ξ ih ),
and (·, ·)H is the inner product in H. Then, for all t ∈ [τi, τi + 1), the control u h (t) given by (1.12) and (2.1) is fed as input to system (1.1). Inf luenced by this control, the trajectory of system (1.11) passes from the state xh(τi) to xh(τi + 1) = xh(τi + 1; τi, xh(τi), u τhi ,τi +1 (·)). Similar operations are executed at the next, (i +1)th step. The method for choosing the control u h (t) on [τi, τi + 1) as given by formula (2.1) is an analogue of the well-known extremal shift method (see [13]). Accordingly, algorithm (1.12), (2.1) for generating control values in system (1.1) can naturally be referred to as an extremal shift algorithm. The solution of the tracking problem for reference motion is given by the following assertion. Theorem 1. Let
x 0 − w0 H ≤ ν 0h.
(2.2)
Let (w(·), ξh(·), ψh(·), xh(·), uh(·)) be a controllable xh-process. Then, for any ε > 0, there are positive h* ∈ (0, 1) such that, for all h ∈ (0, h*), any trajectory xh(·) of system (1.1) generated by the control u h (·) given by (1.12) and (2.1) satisfies the inequality
sup | A −1( x h(τ h,i ) − w(τ h,i ))| H < ε.
τ h,i ∈Δ h
h(·)
be an h-trajectory of system (1.1). By definition, xh(·) corresponds to a control Proof. Let x = x u(·) = u h (·) of form (1.12), (2.1). Let
μ(t ) = | A −1( x h(t ) − w(t ))| 2H ,
t ∈T.
(2.3)
By virtue of inequality (2.2), we obtain
μ(0) ≤ d 0(ν 0h ) 2,
d 0 = d 0( A).
(2.4) h(·)
Consider an arbitrary i = 0, 1, …. By virtue of the semigroup property, representation (1.11) for x similar representation (1.3) for w(·) imply −1
−1
and
A ( x (τ i +1) − w(τ i +1)) = A S (δ i )( x (τ i ) − w(τ i ) h
h
δ
∫
− S (δ i − τ)D(u h(τ i + τ) − u*(τ i + τ))d τ,
δ i = δ i (h),
τ i = τ h,i .
0
Therefore,
μ(τ i +1) = pi
2 H
+ β ih1 + β th2,
(2.5)
where
pi = − A −1S (δ i )( x h(τ i ) − w(τ i )), COMPUTATIONAL MATHEMATICS AND MATHEMATICAL PHYSICS
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β i21
⎛ δi ⎞ = 2 ⎜ pi , S (δ i − τ)D(u h(τ i + τ) − u*(τ i + τ))d τ ⎟ , ⎜ ⎟ ⎝ 0 ⎠H
∫
2
δi 2 βi2
∫ S(δ
=
− τ)D(u (τ i + τ) − u*(τ i + τ))d τ . h
i
0
H
Since {S(t): t ≥ 0} is a contraction semigroup and the operators A–1 and S(δ) commute, we have (2.6) pi H ≤ | A −1( x h(τ i ) − w(τ i ))| 2H = μ(τ i ). Since {S(t): t ≥ 0} is a contraction semigroup, the operator D is bounded, and the set P, which contains the control values u h (·) and u*(·), is bounded as well, we obtain 2
(2.7) β ih2 ≤ d1δ i2, where d1 is a constant depending only on {S(t): t ≥ 0}, D, and P. Let us estimate from above the quantity δi
β ih1
= − 2 ( A −1S (δ i )( x h(τ i ) − w(τ i )), S (δ i − τ)D(u h(τ i + τ) − u*(τ i + τ))) H d τ.
∫ 0
For i = 0, taking into account (2.2), we obtain h (2.8) β 01 ≤ d 2δ 0ν 0h, d 2 = d 2 ( A). Let i ≥ 1. Since the operators of {S(t): t ≥ 0} are self-adjoint and commute with A–1 and in view of the semigroup property, we conclude that
δi
β ih1
= − 2 (S (δ i − τ)A −1S (δ i )( x h(τ i ) − w(τ i )), D(u h(τ i + τ) − u*(τ i + τ))) H d τ
∫ 0
(2.9)
δi
= −2
∫
( pih(τ),
D(u (τ i + τ) − u*(τ i + τ))) H d τ, h
0
where
pih(τ) = − A −1S (2δ i − τ)( x h(τ i ) − w(τ i )). Using relations (1.2) and (1.3) yields | pih(τ) = qih(τ)| H ≤ 2ν ih | A −1| L(H ;H ) sup{| S (2δ i − τ)| L(H ;H ) : τ ∈ [0,2δ i ], Combining (2.9) with this inequality, we derive the estimate
i = 0,1,…} ≤ d3ν ih,
τ ∈ [τ i , τ i +1).
δi
β ih1
∫
≤ 2 (qih(τ), D(u h(τ i + τ) − u*(τ i + τ))) H d τ + d 4ν ihδ i ,
(2.10)
0
where d4 is a constant depending only on D and P. According to the extremal shift algorithm (see (2.1)), the first term on the right-hand side of inequality (2.10) does not exceed d ν ihδ i . Thus, for arbitrary i = 1, 2, …, we have
β ih1 ≤ (d + d 4 )ν ihδ i . Combining (2.5)–(2.8) and (2.11), we transform (2.5) into
(2.11)
(2.12) μ(τ i +1) ≤ μ(τ i ) + d 5{ν ih + δ i }δ i , where d5 is a constant depending only on {S(t): t ≥ 0}, A, D, and P and the function μ(·) is defined by (2.3). Since this inequality holds for all i = 0, 1, …, by virtue of Conditions 1 and 2, we obtain
μ(τ i ) ≤ d 6(ϕ1(h) + ϕ 2(h)), which holds for all i = 0, 1, … . This inequality implies the assertion of the theorem. COMPUTATIONAL MATHEMATICS AND MATHEMATICAL PHYSICS
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Remark 1. The proof of the theorem shows that h* can be written out in explicit form. Theorem 1 implies the following result. Theorem 2. The feedback 9(·, ·, ·) (see (1.6)) given by (2.1) is tracking. Its error estimate has the form
γ V (h) = c1(φ1(h) + φ 2(h))1/2, where c1 is a constant, which can be written out in explicit form. Remark 2. An analysis of the proof of Theorem 1 suggests that Theorems 1 and 2 remain valid if the operators S(t), t ≥ 0, are not self-adjoint. However, the feedback 9 has to be defined as τ i +1 ⎧ ⎪ h = ⎨v τi ,τi +1 ∈ Pτi ,τi +1 (⋅) : ( A −1S (δ i )(ψ ih − ξ ih ), S (τ i +1 − t )Dv h(t )) H dt ⎪⎩ τi τ i +1 ⎧ ⎫ ⎫ ⎪ ⎪ h h h ⎪ −1 ≤ inf ⎨ ( A S (δ i )(ψ i − ξ i ), S (τ i +1 − t )Dv(t )) H dt : v(⋅) τi ,τi +1 ∈ P (⋅) τi ,τi +1 ⎬ + d ν i δ i ⎬ . ⎪⎩ τi ⎪⎭ ⎪⎭
∫
9(τ i , ξ ih, ψ ih )
∫
Let us introduce another condition. Condition 3. There exists a number K0 > 0 such that, for an arbitrary control u(·) ∈ L∞(T; U) taking values in P, the corresponding motion x(·) = x(·; 0, x0, u(·)) of system (1.1) satisfies the estimate |x(t)|H ≤ K0 for all t ∈ T. Moreover, the reference motion satisfies a similar estimate, i.e., |w(t)|H ≤ K* for all t ∈ T, where K* = const > 0. Lemma 1. Let Condition 3 be satisfied. Then the Lipschitz condition |A–1(x(t + ν) – x(t))|H ≤ Kν holds for all t, t + ν∈ T. Proof. Let x(·) be the motion of system (1.1) corresponding to an arbitrary control u(·) ∈ L∞(T; U) with values in P. By virtue of representation (1.1), we have
A −1[ x(t + ν) − x(t )] = β1(t, ν) + β 2(t, ν),
t, t + ν ∈ T ,
ν ≥ 0,
(2.14)
where
β1(t, ν) = A −1[S (ν)x(t ) − x(t )], ν
∫
β 2(t, ν) = S (τ)Du(t + τ)d τ. 0
Note that, according to [14, p. 314], ν
∫
S (ν)z − z = A S (t − τ)zd τ,
ν ≥ 0,
z ∈ H.
0
Since {S(t): t ≥ 0} is a contraction semigroup, the last relation yields the inequality
| A −1{S (ν)z − z}| H ≤ | z| H ν,
ν ≥ 0,
z ∈ H.
(2.15)
Using (2.15) and Condition 3, we obtain
β1(t, ν) H ≤ K 0ν,
t, t + ν ∈ T ,
ν ≥ 0.
(2.16)
In turn, since the operators S(τ) (τ ≥ 0) are contractions, A–1 ∈ +(H; H), and the operator D and the set P are bounded, we conclude that
β 2(t, ν) H ≤ K 1ν,
t, t + ν ∈ T ,
ν ≥ 0,
(2.17)
where K1 > 0 is a number independent of x(·). Comparing (2.14)–(2.17) yields the assertion of the lemma. Note that the constant K can be written out explicitly. The lemma is proved. COMPUTATIONAL MATHEMATICS AND MATHEMATICAL PHYSICS
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Theorem 3. Let Conditions 1–3 be satisfied. Then the assertions of Theorems 1 and 2 remain valid if the feedback 9 (see (1.6)) is given by the formula
{
9(τ i , ξ ih, ψ ih ) = v τhi ,τi +1 (⋅) : v h(t ) = v
t ∈ [τ i , τ i +1),( A −1(ψ ih − ξ ih ), Dv ) H
fo r
}
≤ inf{( A −1(ψ ih − ξ ih ), Dv ) H : v ∈ P } + d ν ih . This theorem is easy to verify by applying Lemma 1 and the inequality −1
−1
A S (2δ i − t )( x (τ i ) − w(τ i )) − A ( x (τ i ) − w(τ i )) h
h
≤ d δi, (1)
H
d
(1)
= const > 0,
(2.18)
which is valid for all t ∈ [0, 2δi] and i = 0, 1, … . Consider the case of P given by m
P =
∑ω u , j
j
ω j ∈U,
u j ∈ R,
(2.19)
j =1
where u = {u1, …, um} ∈ P1 ⊂ Rm, P1 is a bounded closed set, ωj are given elements of U, and the control u(·) on the right-hand side of (1.1) has the following structure: m
u(t ) =
∑ ω u (t). j
j
j =1
Therefore, m
u*(t ) =
∑ ω u*(t). j
j
j =1
In this case, it is natural to choose u h (·) of the same structure. Specifically, we set m
9(τ i , ξ ih, ψ ih )
=
uih(t )
=
∑ u
h ji ω j
for a.e.
t ∈ [τ i , τ i +1),
(2.20)
j =1
where m
∑ u
−1 h h ji ( A (ψ i
− ξ ih ), Dω j ) H
j =1
(2.21)
⎧⎪ m h −1 h ⎫⎪ ≤ inf ⎨ v i ( A (ψ i − ξ ih ), Dω j ) H : v h = {v 1h, … , v mh } ∈ P1 ⎬ + d ν ih. ⎩⎪ j =1 ⎭⎪
∑
Theorem 4. Let Conditions 1–3 be satisfied and the admissible feedback V (1.6) be given by (2.20) and (2.21). Then the assertions of Theorems 1 and 2 hold. The proof of this theorem is similar to that of Theorem 1 and makes use of the equality δi
∫
−1
φ i ≡ (A S (δ i )(ξ i − ψ i ), S (δ i − τ)D(ui (τ i + τ) − u*(τ i + τ))) H d τ h
h
h
0
δi m
=
∫ ∑ (u
h ji
− u*j (τ i + τ))(A −1S (2δ i − τ)(ξ ih − ψ ih ), Dω j ) H d τ.
0 j =1
This relation implies the estimate (see (2.18)) δi m
φi −
∫ ∑ (u
h ji
− u *j (τ i + τ))A −1(ξ ih − ψ ih ), Dω j ) H d τ ≤ d (2)δ i ,
d (2) = const > 0,
i = 1,2,... .
0 j =1
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3. SOLUTION ALGORITHM FOR THE PROBLEM OF TRACKING A REFERENCE CONTROL Let us describe an algorithm for solving the problem of tracking a reference control. Let u*(·) ∈ L∞(T; U) be a reference control taking values in a given convex, bounded, and closed set P ⊂ U, and let w(·) be the reference motion of system (1.3) corresponding to u*(·). Suppose that α = α(h) : (0, 1) → (0, 1) is a function. Before starting the algorithm, we fix h ∈ (0, 1), a sequence {ν ih}i∞=0 , and a partition Δh = {τ h,i }i∞=0 (see (1.5)). The algorithm consists of similar steps. The following operations are executed at the ith step, which is performed on the time interval [τi, τi + 1), τi = τh, i. First, at the time τi, i = 0, 1, …, when the signal
ξ ih ∈ H on the state of system (1.1) and the signal ψ ih ∈ H on the state of reference system (1.3) are received, we choose uα, h(t) according to rule (1.8), where τ i +1 ⎧ ⎪ h h h h 2 = ⎨uτi ,τi +1 (⋅) ∈ Pτi ,τi +1 (⋅) : {(qi (t − τ i ), Du (t )) H + α| u (t )|U }dt ⎪⎩ τi τ i +1 ⎧ ⎫ ⎫ ⎪ ⎪ ⎪ ≤ inf ⎨ {(qih(t − τ i ), Du(t )) H dt + α| u h(t )|U2 } : uτi ,τi +1 (⋅) ∈ Pτi ,τi +1 (⋅)⎬ + d δ i ν ih ⎬ . ⎪⎩ τi ⎪⎭ ⎪⎭
∫
h h 8 α (τ i , ξ i , ψ i )
(3.1)
∫
Then, for all t ∈ [τi, τi + 1), the control uα, h(t) given by (1.8) and (3.1) is fed as input to system (1.1). Influenced by this control, the trajectory of system (1.7) passes from the state xα, h(τi) to the state xα, h(τi + 1) = xα, h(τi + 1; τi, xα, h(τi), uταi,,hτi +1 (·)). Similar operations are executed at the next, (i + 1)th step. This algorithm is the same as algorithm (1.12), (2.1) for solving the problem of tracking reference motion with the only deference being that the control uα, h(·) on [τi, τi + 1) is specified relying on the modified extremal shift rule (3.1). This modification is obtained from the original extremal shift rule by adding 2 the quadratic term α v(t ) U to the minimization functional on the right-hand side of (2.1). In the theory of ill-posed problems (see [15]), this modification corresponds to Tikhonov regularization. Accordingly, algorithm (1.8), (3.1) for generating boundary control values can naturally be called a regularized extremal shift algorithm. The solution of the tracking problem for a reference control is given by the following assertion. Theorem 5. Let Conditions 1 and 2 and inequality (2.2) be satisfied. Suppose that (w(·), ξh(·), ψh(·), uα, h(·)) is a controllable xα, h-process corresponding to feedback (1.6), (3.1) and admissible measurex ments of accuracy h. Then inequalities (1.9) and (1.10) are valid. α, h(·),
Proof sketch. By definition, the trajectory xα, h(·) is generated by a control uα, h(·) satisfying conditions (1.8) and (3.1), where ξ ih ∈ H and ψ ih ∈ H obey inequalities (1.2) and (1.4), i = 1, … . Let t −1
μ α (t ) = | A ( x
α,h
(t ) −
w(t ))| 2H
∫
+ α(h) | u
t α,h
(t )|U2 dt
∫
− α(h) | u*(t )|U2 dt,
0
t ∈T.
0
Following the proof sketch of Theorem 1, we derive the estimate
μ α (τ h,i +1) ≤ μ α (τ h,i ) + c1(δ i + ν ih )δ i
i = 0, 1, … ,
where δi = δi(h) and c1 is a constant independent of h, δ, and α. From this, taking into account the form of the function μα(·), we obtain
| A −1( x α,h(τ h,i ) − w(τ h,i ))| 2H ≤ c1(φ1(h) + φ 2 (h)) + α(h)c3τ h,i , COMPUTATIONAL MATHEMATICS AND MATHEMATICAL PHYSICS
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where c3 = sup{ u U : u ∈ P}. Moreover, it follows from (3.2) that 2
τ h,i
τ h,i
∫ |u
α,h
(t )|U2 dt
≤
0
∫ u*(t)
2 U
dt +
0
c1(φ1(h) + φ 2(h)) . α(h)
(3.3)
By Theorem 5, taking into account (3.2) and (3.3), we conclude the following result. Theorem 6. Let the conditions of Theorem 5 hold. Let also α(h) → 0, {ϕ1(h) + ϕ2(h)} α–1(h) → 0 as h → 0. Then the feedback 8α(·, ·, ·) given by (1.6) and (3.1) is reconstructing. Moreover, the error estimates γU(·) and γ(·) have the form
γU (h) = max{φ1(h) + φ 2(h), α(h)},
γ 1(h) = (φ1(h) + φ 2(h))α −1(h).
Theorem 7. Let the conditions of Theorem 6 and Condition 3 hold. Then the assertions of Theorems 5 and 6 remain valid if the feedback 8α (see (1.6)) is given by the formula
{
8 α (τ i , ξ ih, ψ ih ) = uτhi ,τi +1 (⋅) : u h(t ) = v
for a.e .
t ∈ [τ i , τ i +1),( A −1(ψ ih − ξ ih ), Dv ) H + α| v | 2H
}
≤ inf{( A −1(ψ ih − ξ ih ), Dv ) H + α v H : v ∈ P } + d ν ih . 2
Suppose that the set P has the form of (2.19), where u = {u1, …, um} ∈ P1 ⊂ Rm and P1 is a convex, bounded, and closed set. Then it is natural to set m
8 α (τ i , ξ ih, ψ ih ) = uiα,h(t ) =
∑u
α,h ji ω j
for a.e.
t ∈ [τ i , τ i +1),
(3.4)
j =1
where m
∑
m
∑ |u
u αji,h( A −1(ψ ih − ξ ih ), Dω j ) H + α
j =1
α,h 2 ji |
j =1
m ⎧⎪ m α,h −1 h ⎫⎪ h ≤ inf ⎨ u j ( A (ψ i − ξ i ), Dω j ) H + α | u αj ,h | 2 : u α,h = {u1α,h, … , umα,h} ∈ P1 ⎬ + d ν ih. j =1 ⎩⎪ j =1 ⎭⎪
∑
(3.5)
∑
Theorem 8. Let the conditions of Theorem 7 be satisfied and the admissible feedback 8α (see (1.6)) be given by (3.4) and (3.5). Then the assertions of Theorems 5 and 6 hold. ACKNOWLEDGMENTS This work was supported by the Russian Science Foundation, project no. 14-01-00539. REFERENCES 1. I. Lasiecka and R. Triggiani, “Differential and algebraic Riccati equations with applications to boundary/point control problems: continuous theory and approximation theory,” Lecture Notes in Control and Information Sciences (Springer-Verlag, New York, 1991), Vol. 164. 2. I. Lasiecka, “Unified theory for abstract parabolic boundary problems—a semigroup approach,” Appl. Math. Optim. 6 (4), 287–334 (1980). 3. I. Lasiecka and R. Triggiani, “Dirichlet boundary control problem for parabolic equations with quadratic cost: Analyticity and Riccati feedback synthesis,” SIAM J. Control Optim. 21 (5), 41–67 (1983). 4. F. Flandoli, I. Lasiecka, and R. Triggiani, “Algebraic Riccati equations with nonsmoothing observation arising in hyperbolic and Euler–Bernoulli boundary control problems,” Ann. Mat. Pura Appl. 153, 307–382 (1988). COMPUTATIONAL MATHEMATICS AND MATHEMATICAL PHYSICS
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5. Yu. S. Osipov and A. V. Kryazhimskii, Inverse Problems for Ordinary Differential Equations: Dynamical Solutions (Gordon and Breach, London, 1995). 6. Yu. S. Osipov, L. Pandolfi, and V. I. Maksimov, “The robust boundary control problem: The case of Dirichlet boundary conditions,” Dokl. Math. 62 (2), 205–207 (2000). 7. Yu. S. Osipov, L. Pandolfi, and V. I. Maksimov, “Problems of dynamic reconstruction and robust boundary control: The case of Dirichlet boundary conditions,” J. Inverse III-Posed Probl. 9 (2), 149–162 (2001). 8. V. I. Maksimov, “Reconstruction of controls in exponentially stable linear systems subjected to small disturbances,” J. Appl. Math. Mech. 71 (6), 851–900 (2007). 9. A. V. Kryazhimskiy and V. I. Maksimov, “Resource-saving tracking problem with infinite time horizon,” Differ. Equations 47 (7), 1004–1013 (2011). 10. I. Lasiecka and R. Triggiani, Control Theory for Partial Differential Equations: Continuous and Approximation Theories (Cambridge Univ. Press, Cambridge, 2000), Vol. 1. 11. T. Bretten and K. Kunisch, “Riccati-based feedback control of the monodomain equations with the Fitzhugh– Nagumo model,” SIAM J. Control Optim. 52 (6), 4057–4081 (2014). 12. Y. Wu and X. Xue, “Boundary feedback stabilization of Kirchhoff-type Timoshenko system,” J. Dyn. Control. Syst. 209, 523–538 (2014). 13. N. N. Krasovskii and A. I. Subbotin, Game-Theoretical Control Problems (Nauka, Moscow, 1974; Springer-Verlag, New York, 1988). 14. I. Lasiecka, “Boundary control of parabolic systems: Regularity of optimal solutions,” Appl. Math. Optim. 4 (4), 301–328 (1978). 15. N. Tikhonov and V. Ya. Arsenin, Solutions of Ill-Posed Problems (Halsted, New York, 1978; Nauka, Moscow, 1986).
Translated by I. Ruzanova
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