Optimal control of delay systems with differential and algebraic dynamic constraints

April 2005, Vol. 11, 285–309 DOI: 10.1051/cocv:2005008

OPTIMAL CONTROL OF DELAY SYSTEMS WITH DIFFERENTIAL AND ALGEBRAIC DYNAMIC CONSTRAINTS

Boris S. Mordukhovich

and Lianwen Wang

Abstract. This paper concerns constrained dynamic optimization problems governed by delay con- trol systems whose dynamic constraints are described by both delay-differential inclusions and linear algebraic equations. This is a new class of optimal control systems that, on one hand, may be treated as a specific type of variational problems for neutral functional-differential inclusions while, on the other hand, is related to a special class of differential-algebraic systems with a general delay-differential in- clusion and a linear constraint link between “slow” and “fast” variables. We pursue a twofold goal:

to study variational stability for this class of control systems with respect to discrete approximations and to derive necessary optimality conditions for both delayed differential-algebraic systems under consideration and their finite-difference counterparts using modern tools of variational analysis and generalized differentiation. The authors are not familiar with any results in these directions for such systems even in the delay-free case. In the first part of the paper we establish the value convergence of discrete approximations as well as the strong convergence of optimal arcs in the classical Sobolev space W^1,2. Then using discrete approximations as a vehicle, we derive necessary optimality condi- tions for the initial continuous-time systems in both Euler-Lagrange and Hamiltonian formsviabasic generalized differential constructions of variational analysis.

Mathematics Subject Classification. 49J53, 49K24, 49K25, 49M25, 90C31, 93C30.

Received August 19, 2003. Revised May 3 and June 29, 2004.

1. Introduction

This paper deals with a new class of dynamic optimization problems modelled as follows:

minimize J[x, z] :=ϕ(x(a), x(b)) + _b

a

f(x(t), x(t−∆), z(t),z(t), t) dt˙ (1.1)

Keywords and phrases. Optimal control, variational analysis, functional-differential inclusions of neutral type, differential and algebraic dynamic constraints, discrete approximations, generalized differentiation, necessary optimality conditions.

∗Research was partly supported by the National Science Foundation under grants DMS-0072179 and DMS-0304989.

1 Department of Mathematics, Wayne State University, Detroit, MI 48202, USA.boris@math.wayne.edu

2 Department of Mathematics and Computer Science, Central Missouri State University, Warrensburg, MO 64093, USA;

lwang@cmsu1.cmsu.edu

c EDP Sciences, SMAI 2005

subject to the constraints

z(t)˙ ∈F(x(t), x(t−∆), z(t), t) a.e.t∈[a, b], (1.2)

z(t) =x(t) +Ax(t−∆), t∈[a, b], (1.3)

x(t) =c(t), t∈[a−∆, a), (1.4)

(x(a), x(b))∈Ω⊂IR²ⁿ, (1.5)

wherex: [a−∆, b]→IRⁿ is continuous on [a−∆, a) and [a, b] (with a possible jump att=a), and wherez(t) is absolutely continuous on [a, b]. We always assume that F: IRⁿ×IRⁿ ×IRⁿ×[a, b] →→ IRⁿ is a set-valued mapping of closed graph, that Ω is a closed set, that ∆>0 is a constant delay, and thatA is a constantn×n matrix. Let us label this problem as (P) and note that the methods used in this paper allow us to include the cases of multiple delays ∆₁ ≥∆₂ ≥. . .≥∆_m>0 as well as varying delays ∆(t) with |∆(t)|˙ < α∈(0,1)a.e.

t∈[a, b], which are not consider here for simplicity. We do not also consider the limiting case of ∆↓0 required additional assumptions.

Observe that the variational problem (P) involves two kinds of state variables: “slow” z and “fast” x, which satisfy interrelated dynamic constraints given by the delay-differential inclusion (1.2) and the linear delay-algebraic equation (1.3). Furthermore, the integral functional in (1.1) depends on on both slow and fast variables as well as on thetime-derivative of slow variables (fast variables may not differentiable in time). All these specific features are highly essential for the methods developed and the results obtained in this paper.

On one hand, problem (P) containing both differential and algebraic constraints on slow and fast variables may be viewed as a special subclass of delayeddifferential-algebraic control systems providing, by definition, descriptions of control processvia combinations of interconnected differential and algebraic dynamic relations.

There are many applications of such dynamic models (called DAEs, i.e., differential-algebraic equations) es- pecially in process systems engineering, robotics, mechanical systems with holonomic and nonholonomic con- straints, etc.; see [1, 2, 14, 15] and the references therein. Observe, however, that the dynamic relations (1.2) and (1.3) and the assumptions made are generally different from those conventional in the control theory for DAEs, especially for deriving necessary optimality conditions. The most advanced results in this direction for the so-calledindex oneDAEs are obtained in [14], where it was particularly discovered that optimal processes in such systems donotsatisfy the (strong) Maximum Principle in the absence of a convexity hypothesis on the velocity sets.

On the other hand, the problem (P) under consideration is strongly related to functional-differential control systems of the so-called neutral type, which contain time delays in velocity variables. Indeed, the dynamic constraints (1.2) and (1.3) can be unified as

d dt

x(t) +Ax(t−∆) ∈F

x(t), x(t−∆), x(t) +Ax(t−∆), t a.e.,

that, provided the absolute continuity of ¯x(t) (which is not the case under the assumptions made), may be written in the general form of neutral delay differential inclusions

x(t)˙ ∈G(x(t), x(t−∆),x(t˙ −∆), t) a.e. (1.6)

Similarly, the cost functional (1.1) transfers under this substitution into the form ϕ(x(a), x(b)) +

_b

a

g

x(t), x(t−∆),x(t),˙ x(t˙ −∆), t

dt. (1.7)

Thus we can treat problem (P) as a special case of Bolza-type variational problems for neutral delay-differential inclusions. However, in this way we loose the principal feature of the considered problem (P), which is crucial for the methods applied as well as for the results obtained below. This specific feature of problem (P) is as

follows: both the dynamic constraint (1.6) and the cost functional (1.7) depend in fact not on ˙x(t) and ˙x(t−∆) but on the derivative of the same linear combination x(t) +Ax(t −∆). That is why we treat this linear combination as a new state variable in (1.3) and consider problem (P) in the natural form (1.1)–(1.5), which emphasizes bothdelay-differentialand linear algebraic constraintson the system dynamics.

Our approach is based on the method of discrete approximations, in the line developed in [8, 10–12] for nondelayed differential inclusions, delay-differential inclusions withA= 0, and for a special class of the neutral- type problems that corresponds to (P) withF independent ofzand withf independent of (z,z). Some results˙ for delayed differential-algebraic problems of type (P) were announced in [13] in the case when both F andf are independent ofz, whilef depends on the velocity ˙z described by (1.2).

The discrete approximation method is of undoubted interest from qualitative as well as numerical viewpoints, and the present paper contains results in both of these directions. Our main emphasis, however, is thequalitative aspect, which allows us to derive necessary optimality conditions for delayed differential-algebraic systems by passing to the limit from their discrete-time analogues. A crucial issue is to establish variational stability of discrete approximations that ensures an appropriatestrong convergenceof optimal solutions.

Once such a stability is established, discrete-time control problems for delayeddifference-algebraic inclusions reduce to special finite-dimensional problems of nonsmooth programming with an increasing number of geo- metric constraints that may have empty interiors. To handle such problems, we use appropriate generalized differentiation tools of variational analysis introduced earlier by the first author. In this way we derive necessary optimality conditions for the discrete-time and then for continuous-time problems under consideration.

The rest of the paper is organized as follows. In Section 2 we show that any admissible pair to the delayed differential-algebraic system (1.2) and (1.3) can bestrongly approximatedby the corresponding admissible pairs to its finite-difference counterparts. This result is definitely important for its own sake. It also plays a crucial role in the construction ofwell-poseddiscrete approximations to the original problem (P) and in the subsequent justification of the strong convergence of their optimal solutions to the given optimal solution for (P).

Such a convergence analysis is conducted in Section 3 involving appropriate perturbations of the endpoint constraints (1.5) that is consistent with the step of discretization. The required strong convergence of optimal solutions is justified under an intrinsic property of the original problem (P) calledrelaxation stability. This property imposing the equality between the optimal values in (P) and its relaxation goes far beyond the convexity assumption on the velocity setsF(x, y, z, t).

Section 4 contains basic constructions and required material on generalized differentiation that are appropriate for performing a variational analysis of discrete-time and continuous-time optimal control problems in the subsequent sections. These constructions and calculus rules of generalized differentiation are used in Section 5 for deriving general necessary optimality conditions for nonconvex discrete-time inclusions arising in discrete approximations of the original control problem (P). The main necessary optimality conditions in the forms of Euler-Lagrange and Hamiltonian inclusions for (P) are derived in Section 6 via passing to the limit from discrete approximations.

Our notation is basically standard;cf. [8] and [17]. Recall that, given a set-valued mapping (or multifunction) F: X →→Y between finite-dimensional spaces, thePainlev´e-Kuratowski upper/outer limit ofF(x) as x→x¯ is defined by

Lim sup

x→¯x

F(x) :=

y∈Y| ∃x_k →x,¯ ∃y_k →y with y_k ∈F(x_k) for all k∈IN ,

whereIN stands for the collection of all natural numbers.

2. Discrete approximations of differential-algebraic inclusions

This section deals with discrete approximations of an arbitrary admissible pair to the delayed differential- algebraic system (1.2)–(1.4)withouttaking into account the endpoint constraints. We show that, under fairly general assumptions, any admissible pair to (1.2)–(1.4) can bestrongly approximatedin the sense indicated below by the corresponding admissible pairs to finite-difference inclusions obtained from (1.2)–(1.4) by the classical

Euler scheme. This result isconstructiveprovidingefficient estimates of the approximation rate, and hence it is certainly of independent interest for numerical analysis of delayed differential-algebraic inclusions.

Let (¯x,z) be an¯ admissible pairto (1.2)–(1.4),i.e., ¯x(·) is continuous on [a−∆, a) and [a, b] (with a possible jump at t = a), ¯z(·) is absolutely continuous on [a, b], and relations (1.2)–(1.4) are satisfied. Note that the endpoint constraints (1.5) may not hold for (¯x,¯z); if they do hold, (¯x,z) is called¯ feasibleto (P). The following standing assumptions are imposed throughout the paper:

(H1): There are two open setsU ⊂IRⁿ,V ⊂IRⁿ and two positive numbers_F, m_F such that ¯x(t)∈U for allt∈[a−∆, b] and ¯z(t)∈V for allt∈[a, b], that the setsF(x, y, z, t) are closed, and that one has

F(x, y, z, t)⊂m_FIB for all (x, y, z, t)∈U×U×V ×[a, b], F(x₁, y₁, z₁, t)⊂F(x₂, y₂, z₂, t) +_F(|x₁−x₂|+|y₁−y₂|+|z₁−z₂|)IB

if (x₁, y₁, z₁),(x₂, y₂, z₂)∈U ×U×V andt∈[a, b], whereIB stands for the closed unit ball inIRⁿ. (H2): F(x, y, z, t) is Hausdorff continuous fora.e.t∈[a, b] uniformly in (x, y, z)∈U×U×V. (H3): The functionc(·) is continuous on [a−∆, a].

Following [3], we consider the so-calledaveraged modulus of continuity for the multifunction F(x, y, z, t) with (x, y, z)∈U×U ×V andt∈[a, b] defined by

τ(F;h) :=

_b

a

σ(F;t, h) dt, whereσ(F;t, h) := sup

ϑ(F;x, y, z, t, h)(x, y, z)∈U×U×V with ϑ(F;x, y, z, t, h) := sup haus(F(x, y, z, t₁), F(x, y, z, t₂))(t₁, t₂)∈

t−h

2, t+h 2

∩[a, b]

,

and where haus(·,·) stands for the Hausdorff distance between two compact sets. It is proved in [3] that if F(x, y, z, t) is Hausdorff continuous for a.e. t∈[a, b] uniformly in (x, y, z)∈U×U ×V, thenτ(F;h)→0 as h→0. This fact is essentially used in what follows.

Let us construct a sequence of discrete approximations of the given delayed differential-algebraic inclusion replacing the derivative in (1.2) by the classicalEuler finite difference

z(t)˙ ≈z(t+h)−z(t)

h ·

For any N ∈ IN := {1,2, . . .} we consider the step of discretization h_N := ∆/N and define the discrete partition t_j := a+jh_N as j = −N, . . . , k and t_k+1 := b, where k is a natural number determined from a+kh_N ≤ b < a+ (k + 1)h_N. Then the corresponding delayed difference-algebraic inclusions associated with (1.2)–(1.4) are described by







z_N(t_j+1)∈z_N(t_j) +h_NF(x_N(t_j), x_N(t_j−∆), z_N(t_j), t_j) for j= 0, . . . , k, z_N(t_j) =x_N(t_j) +Ax_N(t_j−∆) for j= 0, . . . , k+ 1,

x_N(t_j) =c(t_j) for j=−N, . . . ,−1.

(2.1)

Given discrete functionsx_N(t_j) andz_N(t_j) satisfying (2.1), we consider theextensionofx_N(t_j) to the continuous- time intervals [a−∆, b] such thatx_N(t) are defined piecewise-linearly on [a, b] and piecewise-constantly, contin- uously from the right on [a−∆, a). We also define piecewise-constantextension of discrete velocitieson [a, b] by

v_N(t) :=z_N(t_j+1)−z_N(t_j)

h_N , t∈[t_j, t_j+1), j= 0, . . . , k.

It is easy to see that

z_N(t) =z_N(a) + _t

a

v_N(s) ds fort∈[a, b], wherez_N(t) =x_N(t) +Ax_N(t−∆).

LetW^1,2[a, b] be the classical Sobolev space with the norm x(·)_W^1,2 := max

t∈[a,b]|x(t)|+ b

a

|x(t)|˙ ²dt _1/2

.

The following theorem, which plays an essential role in the subsequent constructions and results of the paper being also important for its own sake, establishes the strongW^1,2-approximation of any admissible pair to the given delayed differential-algebraic inclusion by corresponding solutions to its discrete-time counterparts.

Theorem 2.1. Let (¯x,z)¯ be an admissible pair to (1.2)–(1.4) under hypotheses (H1)–(H3). Then there is a sequence (x_N(t_j),z_N(t_j))of solutions to discrete inclusions (2.1) withx_N(t₀) = ¯x(a) for all N ∈IN such that their extensionsx_N(t), a−∆≤t≤b, converge uniformly tox(·)¯ on[a−∆, b]whilez_N(t), a≤t≤b, converge toz(t)¯ in theW^1,2-norm on[a, b] asN → ∞.

Proof. Using the density of step-functions in L¹[a, b], we first select a sequence {ω_N(·)}, N ∈ IN, such that eachω_N(t) is constant on the interval [t_j, t_j+1) forj = 0, . . . , k and thatω_N(·) converge to ˙¯z(·) as N → ∞in the norm topology ofL¹[a, b]. It follows from (H1) that

|ω_N(t)| ≤ω_N(t)−z(t)˙¯ +z(t)˙¯ ≤1 +m_F for allt∈[a, b] andN ∈IN. In the estimates below we use the sequence

ξ_N :=

_b

a

ω_N(t)−z(t)˙¯ dt→0 as N → ∞. Denoteω_Nj :=ω_N(t_j) and define discrete pairs (u_N(t_j), s_N(t_j)) recurrently by







u_N(t_j) := ¯x(t_j) for j=−N, . . . ,0,

s_N(t_j) :=u_N(t_j) +Au_N(t_j−∆) forj= 0, . . . , k+ 1, s_N(t_j+1) :=s_N(t_j) +h_Nω_Nj forj= 0, . . . , k.

Then the extended discrete functions satisfy











u_N(t) = ¯x(t_j) fort∈[t_j, t_j+1), j=−N, . . . ,−1, s_N(t) =u_N(t) +Au_N(t−∆) fort∈[a, b], s_N(t) = ¯z(a) +

_t

a

ω_N(s) ds fort∈[a, b].

Next we want to prove that u_N(t) converge uniformly to ¯x(t) on [a, b]. Denote r_N(t) := u_N(t)−x(t) and¯ y_N(t) :=|r_N(t) +Ar_N(t−∆)|. For anyt∈[a, b] one has

y_N(t) =s_N(t)−z(t)¯ ≤ _t

a

|ω_N(s)−z(s)|˙¯ ds≤ ξ_N, (2.2)

which implies the estimates

|r_N(t)| ≤y_N(t) +|A||r_N(t−∆)| ≤y_N(t) +|A|y_N(t−∆) +|A|²|r_N(t−2∆)| ≤. . .

≤y_N(t) +|A|y_N(t−∆) +. . .+|A|^my_N(t−m∆) +|A|^m+1|r_N(t−(m+ 1)∆)|.

Observe that c(·) is uniformly continuous on [a−∆, a] due to assumption (H3). Picking an arbitrary se- quence β_N ↓0 asN → ∞, we therefore have

|c(t)−c(t)| ≤β_N whenever t, t∈[t_j, t_j+1], j=−N, . . . ,−1.

Choose an integer number m such thata−∆ ≤b−(m+ 1)∆< a. Thent−(m+ 1)∆∈[t_j, t_j+1) for some j∈ {−N, . . . ,−1}, which implies that

|r_N(t−(m+ 1)∆)| ≤ |c(t_j)−c(t−(m+ 1)∆)| ≤β_N. Sincem∈IN does not depend onN, this gives

|r_N(t)| ≤ξ_N(1 +|A|+. . .+|A|^m) +|A|^m+1β_N :=_N →0 asN → ∞ (2.3) for allt∈[a, b] due to the construction ofr_N(·). Now consider a sequence{ζ_N} defined by

ζ_N := h_N k j=0

dist(ω_Nj;F(u_N(t_j), u_N(t_j−∆), s_N(t_j), t_j))

and show thatζ_N ↓ 0 asN → ∞. By construction ofζ_N and the averaged modulus of continuityτ(F;h) we get the following estimates:

ζ_N = k j=0

_tj+1

tj

dist(ω_Nj;F(u_N(t_j), u_N(t_j−∆), s_N(t_j), t_j)) dt

= k j=0

_tj+1

tj

dist(ω_Nj;F(u_N(t_j), u_N(t_j−∆), s_N(t_j), t)) dt

+ k j=0

_tj+1

tj

dist(ω_Nj;F(u_N(t_j), u_N(t_j−∆), s_N(t_j), t_j))−dist(ω_Nj;F(u_N(t_j), u_N(t_j−∆), s_N(t_j), t)) dt

≤ k j=0

_tj+1

tj

dist(ω_Nj;F(u_N(t_j), u_N(t_j−∆), s_N(t_j), t)) dt+ k j=0

_tj+1

tj

σ(F;t, h_N) dt

≤ k j=0

_tj+1

tj

dist(ω_Nj;F(u_N(t_j), u_N(t_j−∆), s_N(t_j), t)) dt+τ(F;h_N).

Further, assumption (H1) implies that for anyt∈[t_j, t_j+1) withj= 0, . . . , k one has

dist(ω_Nj;F(u_N(t_j), u_N(t_j−∆), s_N(t_j), t))−dist(ω_Nj;F(u_N(t), u_N(t−∆), s_N(t), t))

≤dist(F(u_N(t_j), u_N(t_j−∆), s_N(t_j), t), F(u_N(t), u_N(t−∆), s_N(t), t))

≤_F(|u_N(t_j)−u_N(t)|+|u_N(t_j−∆)−u_N(t−∆)|+|s_N(t_j)−s_N(t)|).

Taking into account that

s_N(t_j)−s_N(t)=

_t

tj

ω_N(s) ds

≤(1 +m_F)(t_j+1−t_j) = (1 +m_F)h_N :=a_N ↓0, we arrive at

|u_N(t)−u_N(t_j)| ≤a_N+|A||u_N(t−∆)−u_N(t_j−∆)|

≤a_N(1 +|A|+. . .+|A|^m) +|A|^m+1|u_N(t−(m+ 1)∆)−u_N(t_j−(m+ 1)∆)|

≤a_N(1 +|A|+. . .+|A|^m) +|A|^m+1β_N :=b_N ↓0 as N → ∞ and hence ensure that

dist(ω_Nj;F(u_N(t_j), u_N(t_j−∆), s_N(t_j), t))−dist(ω_Nj;F(u_N(t), u_N(t−∆), s_N(t), t))≤(a_N+ 2b_N)_F. It follows from (H1), (2.2) and (2.3) that for anyt∈[t_j, t_j+1) andj= 0, . . . , k one has

dist(ω_Nj;F(u_N(t), u_N(t−∆), s_N(t), t))−dist(ω_N(t);F(¯x(t),x(t¯ −∆),z(t), t))¯

≤dist(F(u_N(t), u_N(t−∆), s_N(t), t), F(¯x(t),x(t¯ −∆),z(t), t))¯

≤_F(|u_N(t)−x(t)|¯ +|u_N(t−∆)−x(t¯ −∆)|+|s_N(t)−¯z(t)|)

≤(2_N +ξ_N)_F.

Combining the above estimates and denotingµ_N :=a_N+ 2b_N + 2_N +ξ_N, we arrive at

dist(ω_Nj;F(u_N(t_j), u_N(t_j−∆), s_N(t_j), t))≤_Fµ_N+ dist(ω_Nj;F(¯x(t),x(t¯ −∆),z(t), t))¯ ≤_Fµ_N +ω_Nj −z(t)˙¯

and finally conclude that ζ_N ≤

k j=0

_tj+1

tj

ω_Nj−z(t)˙¯ +_Fµ_N

dt+τ(F;h_N)

=ξ_N +_Fµ_N(b−a) +τ(F;h_N) :=γ_N ↓0 as N → ∞. (2.4) Note that the discrete functions (u_N(t_j), s_N(t_j)) maynotbe a admissible pair for (2.1) because the inclusions ω_Nj ∈ F(u_N(t_j), u_N(t_j−∆), s_N(t_j), t_j) may not be true for j = 0, . . . , k. Let us construct the desired pair (x_N(t_j),z_N(t_j)) by the followingproximal algorithm:





















x_N(t_j) =c(t_j) for j=−N, . . . ,−1, x_N(t₀) = ¯x(a),

z_N(t_j+1) =z_N(t_j) +h_Nv_Nj for j= 0, . . . , k,

z_N(t_j) =x_N(t_j) +Ax_N(t_j−∆) for j= 0, . . . , k+ 1, v_Nj ∈F(x_N(t_j),x_N(t_j−∆),z_N(t_j), t_j) for j= 0, . . . , k,

|v_Nj−ω_Nj|= dist(ω_Nj;F(x_N(t_j),x_N(t_j−∆),z_N(t_j), t_j)) for j = 0, . . . , k.

(2.5)

It follows from the construction (2.5) that (x_N(t_j),z_N(t_j)) is a feasible pair to the discrete inclusion (2.1) for eachN ∈IN. Note that

|x_N(t)−x(t)¯ |=|x_N(t_j)−x(t)¯ |=|c(t_j)−c(t)|< β_N fort∈[t_j, t_j+1), j=−N, . . . ,−1,

which implies that the extensions ofx_N(·) converge to ¯x(t) uniformly on [a−∆, a). Let us analyze the situation on [a, b].

First we claim that x_N(t_j)∈ U and z_N(t_j)∈ V forj = 0, . . . , k+ 1. Arguing by induction, we obviously havex_N(t₀)∈U andz_N(t₀)∈V. Assume thatx_N(t_j)∈U andz_N(t_j)∈U for allj= 1, . . . , mwith some fixed m∈ {1, . . . , k}. Then

|x_N(t_m+1)−u_N(t_m+1)|=|z_N(t_m+1)−Ax_N(t_m+1−∆)−s_N(t_m+1) +Au_N(t_m+1−∆)|

≤ |A||x_N(t_m+1−∆)−u_N(t_m+1−∆)|+|z_N(t_m+1)−s_N(t_m+1)|

≤ |A||x_N(t_m+1−∆)−u_N(t_m+1−∆)|+|A||x_N(t_m−∆)−u_N(t_m−∆)|

+|x_N(t_m)−u_N(t_m)|+h_Ndist(ω_Nm;F(x_N(t_m),x_N(t_m−∆),z_N(t_m), t_m)).

Taking into account that

|x_N(t_m)−u_N(t_m)| ≤ |A||x_N(t_m−N)−u_N(t_m−N)|+|A||x_N(t_m−1−N)−u_N(t_m−1−N)|

+|x_N(t_m−1)−u_N(t_m−1)|+h_Ndist(ω_Nm−1;F(x_N(t_m−1),x_N(t_m−1−N),z_N(t_m−1), t_m−1)), that

dist(ω_Nm−1;F(x_N(t_m−1),x_N(t_m−1−N),z_N(t_m−1), t_m−1))

≤dist(ω_Nm−1;F(u_N(t_m−1), u_N(t_m−1−N), s_N(t_m−1), t_m−1)) +_F(|x_N(t_m−1)−u_N(t_m−1)|+|z_N(t_m−1)−s_N(t_m−1)|

+|x_N(t_m−1−N)−u_N(t_m−1−N)|), (2.6) that

|z_N(t_m)−s_N(t_m)| ≤ |x_N(t_m)−u_N(t_m)|+|A||x_N(t_m−N)−u_N(t_m−N)|, (2.7) and that |x_N(t_j)−u_N(t_j)|= 0 forj≤0, one has

|x_N(t_m+1)−u_N(t_m+1)| ≤M₁h_N m j=0

dist(ω_Nj;F(u_N(t_j), u_N(t_j−∆), s_N(t_j), t_j))≤M₁γ_N (2.8)

with some constantM₁>0. Now invoking (2.3) and increasingM₁if necessary, we arrive at

|x_N(t_m+1)−x(t¯ _m+1)| ≤ξ_N+M₁γ_N →0 as N → ∞, which implies thatx_N(t_j)∈U forj= 0, . . . , k+ 1.

Observing further that

|z_N(t_m+1)−s_N(t_m+1)| ≤ |z_N(t_m)−s_N(t_m)|+h_N|v_Nm−ω_Nm|

≤ |z_N(t_m)−s_N(t_m)|+h_Ndist(ω_Nm;F(x_N(t_m),x_N(t_m−∆),z_N(t_m), t_m)), we derive from (2.6) and (2.7) the estimate

|z_N(t_m+1)−s_N(t_m+1)| ≤M₂h_N m j=0

dist(ω_Nj;F(u_N(t_j), u_N(t_j−∆), s_N(t_j), t_j))≤M₂γ_N (2.9)

with some constantM₂>0. Note that

|z_N(t_m+1)−z¯_N(t_m+1)| ≤ |z_N(t_m+1)−s_N(t_m+1)|+|s_N(t_m+1)−z¯_N(t_m+1)| ≤M₂γ_N+ξ_N,

which ensures the inclusionsz_N(t_j)∈V forj= 0, . . . , k+ 1. It remains to prove thatz_N(t) converge to ¯z(t) in theW^1,2-norm on [a, b], which means that

t∈[a,b]maxz_N(t)−z(t)¯ + _b

a

z˙_N(t)−z(t)˙¯ ²dt→0 as N → ∞. (2.10)

To furnish this, we use (2.8) and (2.9) to get the estimates

k+1

j=0

x_N(t_j)−u_N(t_j)≤^k+1

j=0

M₁

j−1

m=0

h_Ndist(ω_Nm;F(u_N(t_m), u_N(t_m−∆), s_N(t_m), t_m))

≤M₁(b−a) k j=0

dist(ω_Nj;F(u_N(t_j), u_N(t_j−∆), s_N(t_j), t_j)),

k+1

j=0

z_N(t_j)−s_N(t_j)≤^k+1

j=0

M₂

j−1

m=0

h_Ndist(ω_Nm;F(u_N(t_m), u_N(t_m−∆), s_N(t_m), t_m))

≤M₂(b−a) k j=0

dist(ω_Nj;F(u_N(t_j), u_N(t_j−∆), s_N(t_j), t_j)),

which imply by (H1) that _b

a

z˙_N(t)−ω_N(t)dt= k j=0

_tj+1

tj

z˙_N(t)−ω_N(t)dt

= k j=0

_tj+1

tj

|v_Nj −ω_Nj|dt = k j=0

h_Ndist(ω_Nj;F(x_N(t_j),x_N(t_j−∆),z_N(t_j), t_j))

= k j=0

h_Ndist(ω_Nj;F(u_N(t_j), u_N(t_j−∆), s_N(t_j), t_j))

+ k j=0

h_N[dist(ω_Nj;F(x_N(t_j),x_N(t_j−∆),z_N(t_j), t_j))

−dist(ω_Nj;F(u_N(t_j), u_N(t_j−∆), s_N(t_j), t_j))]

≤^k

j=0

h_Ndist(ω_Nj;F(u_N(t_j), u_N(t_j−∆), s_N(t_j), t_j))

+ k j=0

_Fh_N

|x_N(t_j)−u_N(t_j)|+|x_N(t_j−∆)−u_N(t_j−∆)|+|z_N(t_j)−s_N(t_j)|

≤γ_N + k j=0

_Fh_N

|x_N(t_j)−u_N(t_j)|+|x_N(t_j−∆)−u_N(t_j−∆)|+|z_N(t_j)−s_N(t_j)|

≤γ_N + 2(M₁+M₂)(b−a)_F k j=0

h_Ndist(ω_Nj;F(u_N(t_j), u_N(t_j−∆), s_N(t_j), t_j))

≤γ_N + 2(M₁+M₂)_F(b−a)γ_N.

The latter ensures the estimates _b

a

z˙_N(t)−z(t)˙¯ dt≤ _b

a

z˙_N(t)−ω_N(t)dt+ _b

a

ω_N(t)−z(t)˙¯ dt

≤γ_N(1 + 2(M₁+M₂)(b−a)_F) +ξ_N.

Due to x_N(t)∈U and z_N(t)∈ V, we get from (H1) by (1.2) and (2.5) thatz˙_N(t)≤m_F,z(t)˙¯ ≤m_F, and hence

_b

a

z˙_N(t)−z(t)˙¯ ² dt= _b

a

z˙_N(t)−z(t)˙¯ z˙_N(t) + ˙¯z(t)dt

≤ 2m_F[γ_N(1 + 2(M₁+M₂)(b−a)_F) +ξ_N]↓0 as N → ∞. Observing finally that

t∈[a,b]maxz_N(t)−z(t)¯ ²≤(b−a) _b

a

z˙_N(t)−z(t)˙¯ ²dt,

we arrive at (2.10) and complete the proof of the theorem.

3. Strong convergence of discrete approximations

The goal of this section is to construct a sequence of well-posed discrete approximations of the dynamic optimization problem (P) such that optimal solutions to discrete approximation problemsstrongly converge, in the sense described below, to a given optimal solution to the original optimization problem governed by delayed differential-algebraic inclusions. The following construction explicitly involves the optimal solution (¯x,¯z) to the problem (P) under consideration for which we aim to derive necessary optimality conditions in the subsequent sections.

For any natural number N we consider the followingdiscrete-timedynamic optimization problem (P_N) for functional-difference inclusions:

minimize J_N[x_N, z_N] :=ϕ(x_N(t₀), x_N(t_k+1)) +|x_N(t₀)−x(a)¯ |² +h_N

k j=0

f

x_N(t_j), x_N(t_j−∆), z_N(t_j),z_N(t_j+1)−z_N(t_j)

h_N , t_j

+ k j=0

_tj+1

tj

z_N(t_j+1)−z_N(t_j) h_N −z(t)˙¯

² dt (3.1) subject to thedynamic constraintsgoverned by (2.1), the perturbed endpoint constraints

(x_N(t₀), x_N(t_k+1))∈Ω_N := Ω +η_NIB, (3.2) where η_N := |x_N(t_k+1)−x(b)¯ | with the approximation x_N(t) of ¯x(t) from Theorem 2.1, and the auxiliary constraints

|x_N(t_j)−x(t¯ _j)| ≤ε, |z_N(t_j)−z(t¯ _j)| ≤ε, j= 1, . . . , k+ 1, (3.3) with some ε > 0. The latter auxiliary constraints are needed to guarantee theexistence of optimal solutions in (P_N) and can beignoredin the derivation of necessary optimality conditions; see below.

In what follows we selectε >0 in (3.3) such that ¯x(t) +εIB⊂U for allt∈[a−∆, b] and ¯z(t) +εIB⊂V for all t∈[a, b]. Take sufficiently largeN ensuring thatη_N < ε. Note that problems (P_N) havefeasiblesolutions, since the pair (x_N,z_N) from Theorem 2.1 satisfy all the constraints (2.1), (3.2), and (3.3). Therefore, by the classical Weierstrass theorem, each (P_N) admits an optimal pair (¯x_N,z¯_N) under the following assumption imposed in addition to (H1)–(H3):

(H4): ϕis continuous on U ×U, f(x, y, z, v,·) is continuous fora.e.t ∈[a, b] uniformly in (x, y, z, v)∈ U×U×V ×m_FIB,f(·,·,·,·, t) is continuous onU ×U×V ×m_FIB uniformly in t∈[a, b], and Ω is locally closed around (¯x(a),x(b)).¯

We are going to justify the strong convergence of (¯x_N,z¯_N) to (¯x,z) in the sense of Theorem 2.1. To proceed,¯ we need to involve an important intrinsic property of the original problem (P) calledrelaxation stability. Let us consider, along with the original system (1.2), theconvexifieddelayed differential-algebraic system

z(t)˙ ∈coF(x(t), x(t−∆), z(t), t) a.e.t∈[a, b],

z(t) =x(t) +Ax(t−∆), t∈[a, b], (3.4)

where “co ” stands for the convex hull of a set. Further, given the integrandf in (1.1), we take its restriction f_F(x, y, z, v, t) := f(x, y, z, v, t) +δ(v;F(x, y, z, t))

toFin (1.2), whereδ(·;F) stands for the indicator function of a set. Denote byf_F(x, y, z, v, t) theconvexification of f_F in the v variable and define therelaxed generalized Bolza problem (R) for delayed differential-algebraic systems as follows:

minimizeJ[x, z] :=ϕ(x(a), x(b)) + _b

a

f_F(x(t), x(t−∆), z(t),z(t), t) dt˙ (3.5) over feasible pairs (x, z) with the same analytic properties as in (P) subject to the tail (1.4) and endpoint (1.5) constraints. Every feasible pair to (R) is called arelaxed pairto (P).

One clearly has inf(R)≤ inf(P) for the optimal values of the cost functionals in the relaxed and original problems. We say that the original problem (P) isstable with respect to relaxationif

inf(P) = inf(R).

This property, which obviously holds under the convexity assumptions on the sets F(x, y, z, t) and the inte- grand f in v, goes far beyond the convexity. General sufficient conditions for the relaxation stability of (P) follow from [4]. We also refer the reader to [8, 10, 19, 20] for more detailed discussions on the validity of the relaxation stability property for various classes of differential and functional-differential control systems.

Now we are ready to establish the following strong convergence theorem for optimal solutions to discrete approximations, which makes a bridge between optimal control problems governed by delayed differential- algebraic and difference-algebraic systems.

Theorem 3.1. Let (¯x,¯z) be an optimal pair to problem (P), which is assumed to be stable with respect to relaxation. Suppose also that hypotheses (H1)–(H4) hold. Then any sequence{(¯x_N,z¯_N)},N ∈IN, of optimal pairs to (P_N) extended to the continuous interval [a−∆, b] and [a, b] respectively, strongly converges to (¯x,¯z) asN → ∞ in the sense thatx¯_N converge tox¯ uniformly on [a−∆, b]andz¯_N converge to ¯z in theW^1,2-norm on [a, b].

Proof. We know from the above discussion that (P_N) has an optimal pair (¯x_N,z¯_N) for allN sufficiently large;

suppose that it happens for all N ∈ IN without loss of generality. We consider the sequence (x_N,z_N) from Theorem 2.1. Since each (x_N,z_N) is feasible to (P_N), one has

J_N[¯x_N,z¯_N]≤J_N[x_N,z_N] for all N ∈IN.