\begin{matrix}c \\ ◯ \end{matrix}Birkhäuser Verlag Basel 1

Multiobjective optimal control problems

Christiane Tammer ^{∗}

Abstract. From an Ekeland-type variational principle for vector optimization problems we derive an ϵ-minimum principle in the sense of Pontrjagin for suboptimal controls using classical results for differential equations.

Key words: Multicriteria control problems, ϵ-variational inequalities, variational principle Mathematics Subject Classification: 49 J 15, 49 J 40, 90 C 29, 65 K 10

1. Introduction

In our present paper we study suboptimal controls of a class of multiobjective optimal control problems. In control theory often one has the problem to minimize more than one objective function, for instance a cost functional as well as the distance between the final state and a given point. To realize this task usually one takes as objective function a weighted sum of the different objectives. However, the more natural way would be to study the set of efficient points of a vector optimization problem with the given objective functions. It is well known that the weighted sum is only a special surrogate problem to find efficient points, which has the disadvantage that in the nonconvex case one can not find all efficient elements on this way. Necessary conditions for solutions of multiobjective dynamic programming or control problems were derived by several authors, see Klötzler [14], Benker and Kossert [2], Breckner [3], Gorochowik and Kirillowa [9], Gorochowik [8], [10] and Salukvadze [19]. It is difficult to show the existence of an optimal control (see Klötzler [15]), whereas suboptimal controls exist under very weak assumptions. So it is important to derive some assertions for suboptimal controls. Ekeland’s variational principle [5], [6], [1] is a very deep result in optimization theory and says that there exists an exact solution of a slightly perturbed optimization problem in a neighbourhood of an approximate solution of the original problem. This principle can be used in order to derive necessary conditions for solutions and approximate solutions of optimization problems. Many authors have published extensions and applications of Ekeland’s variational principle as well as equivalent statements. Extensions of Ekeland’s variational principle to vector optimization were given by several authors. Loridan (1984) [16] has presented a vector-valued variational principle

∗Martin-Luther-University Halle-Wittenberg, Department of Mathematics and Informatics, D-06099 Halle, Germany

for the finite-dimensional case where he could apply Ekeland’s original result directly by taking a suitable scalarization. Vector-valued variational principles for an objective function, which take its values in general spaces have been derived by Nemeth (1986) [17], Khanh (1986) [13], Tammer (1992) [20], [23], Dentscheva and Helbig (1996) [4] and Isac (1996) [12]. In our papers [22] and [11] we have used the vector-valued variational principle of [20] in order to show ϵ-variational inequalities (or ϵ-Kolmogorov conditions) for approximate solutions of a general approximation problem. The aim of this paper is to derive an ϵ-minimum principle for suboptimal controls of multicriteria control problems from the vector-valued variational principle.

2. Solution concepts in multicriteria optimization

We formulate a multicriteria control problem with an objective function which takes it values in the m-dimensionl Euclidean space R^{m}. In the following we denote the topological interior of a set C ⊂ R^{m} by int C, the topological boundary of C by bd C and the topological closure of C by cl C. Let us assume:

(A1): (V, d) is a complete metric space, K ⊂ R^{m} is a convex cone with k^{0} ∈ int K , B ⊂ R^{m} is a pointed convex cone with int B ≠ ∅ such that cl B + (K \ {0}) ⊂ int B.

(A2): F : V ⟶ R^{m} is lower semicontinuous with respect to k^{0} and B in the sense that M_{r} = { v ∈ V | F (v) ∈ rk^{0} − cl B } is closed for each r ∈ R and bounded from below, i. e., F [V] ⊂ y + B for a certain y ∈ R^{m}.

Now we consider the following vector optimization problem to determine the efficient point set of F [V] with respect to K :

(P) : Compute the set Eff(F [V], K),

where

Eff(F [V], K) = { F (v̄) | v̄ ∈ V and F [V] ∩ (F (v̄) − (K \ {0})) = ∅ }.

Furthermore, we will introduce approximately efficient elements of vector optimization problems. The reason for introducing approximately efficient solutions is the fact that numerical algorithms usually generate only approximative solutions anyhow and moreover that the efficient point set may be empty in the general noncompact case, whereas approximately efficient points always exist under very weak assumptions (see Tammer [21], where existence results for approximate solutions of a vector optimization problem were shown, especially under the assumption that the objective function is bounded from below).

Definition 1 An element F (v_{ϵ}) ∈ F [V] is called an approximately efficient point of F [V] with respect to K, k^{0} ∈ int K and ϵ > 0, if

F [V] ∩ (F (v_{ϵ}) − ϵk^{0} − (K \ {0})) = ∅.

The approximately efficient point set of F [V] with respect to K, k^{0} and ϵ is denoted by Eff(F [V], K_{ϵk^{0}}), where K_{ϵk^{0}} := ϵk^{0} + K . Moreover, we will study approximately efficient elements with respect to the cone B from assumption (A1) instead of K.

3. Variational principles in partially ordered spaces

Beginning with the paper of Loridan [16] various authors proved variational principles of Ekeland’s type for ϵ-efficient solutions of multicriteria optimization problems by scalarization. In [20] we have shown a vector-valued variational principle by using a separation theorem for nonconvex sets and Ekeland’s original result. Here we will use the following variational principle which was shown in [23] and which is a sharper result than in [20].

The following assertion can be considered as an extension of Ekeland’s variational principle ([5], [6]) to vector optimization.

Theorem 1 ([23]) Assume (A1) and (A2). Then for any ϵ > 0, λ > 0 and any F (v_{0}) ∈ Eff(F [V], B_{ϵk^{0}}) there exists an element v_{ϵ} ∈ V such that

1. F (v_{ϵ}) ∈ f(v_{0}) − λd(v_{0}, v_{ϵ})k^{0} − cl B and F (v_{ϵ}) ∈ Eff(F [V], D_{ϵk^{0}}), where D is an open subset of R^{m} with K \ {0} ⊂ D, 0 ∈ bd D and cl D + (K \ {0}) ⊂ D,

2. d(v_{0}, v_{ϵ}) ≤ ϵ/λ,

3. F_{λk^{0}}(v_{ϵ}) ∈ Eff(F_{λk^{0}}[V], K)

are fulfilled, where F_{λk^{0}}(v) := F (v) + λd(v, v_{ϵ}) k^{0}.

The following theorem follows immediately from Theorem 1 regarding the fact that under the given assumptions there always exists an approximately efficient element v_{0} ∈ V with

F (v_{0}) ∈ Eff(F [V], B_{ϵk^{0}})

(cf. Tammer [21]).

Theorem 2 Assume (A1), (A2). Then for every ϵ > 0 there exists some point v_{ϵ} ∈ V such that

1. F [V] ∩ (F (v_{ϵ}) − ϵk^{0} − (K \ {0})) = ∅,

2. F_{ϵk^{0}}[V] ∩ (F_{ϵk^{0}}(v_{ϵ}) − (K \ {0})) = ∅ , where F_{ϵk^{0}}(v) := F (v) + d(v, v_{ϵ})ϵk^{0}.

Remark 1: Theorem 1 is slightly stronger than Theorem 2. The main difference concerns condition (ii) in Theorem 1, which gives the whereabouts of point x_{ϵ} in V.

Remark 2: The main result of the last theorems (statement 3 Theorem 1 and statement 2 in Theorem 2) says that v_{ϵ} is an efficient solution of a slightly perturbed vector optimization problem. This statement can be used in order to derive necessary conditions for approximately efficient elements. In our papers [22] and [11] we have shown ϵ - Kolmogorov - conditions for approximately efficient solutions of abstract approximation problems applying the third condition in Theorem 1. In the next chapter we will use Theorem 2 in order to derive an ϵ- minimum - principle in the sense of Pontrjagin for suboptimal solutions of multicriteria control problems.

4. An ϵ - minimum principle for multiobjective optimal control problems

In this chapter we will give an application of the multicriteria variational principle in Theorem 2 to control problems.

Consider the system of differential equations

\begin{matrix}\frac{dx}{dt}(t) = & φ(t, x(t), u(t)), \\ x(0) = & x_{0} ∈ R^{n} \end{matrix} \}  (1)

and the control restriction

u(t) ∈ U,

which must hold almost everywhere on [0, T] with T > 0. We assume that

(C1) φ : [0, T] × R^{n} × U ⟶ R^{n} is continuous and U is compact.

The vector x(t) describes the state of the system, u(t) is the control at time t and belongs to the set U.

Furthermore, suppose that

(C2) \frac{∂φ}{∂x_{i}}, i=1,...,n, are continuous on [0, T] × R^{n} × U, (C3) (x, φ(t, x, u)) ≤ c(1 + ||x||^{2}) for some c > 0.

Remark 3: Let be u : [0, T] ⟶ U a measurable control. Condition (C2) and the continuity of φ ensure that there exists a unique solution x of the differential equation (1) on [0, τ] for a sufficiently small τ > 0. By using Gronwall’s inequality condition (C3) implies

||x(t)||^{2} ≤ (||x_{0}||^{2} + 2cT)e^{2cT} ,

and, hence, ensures the existence of the solution on the whole time interval [0, T]. Moreover, the last inequality yields

||\frac{dx(t)}{dt}|| ≤ max\{φ(t, x, u) | (t, x, u) ∈ [0, T] × B^{0} × U\},

where B^{0} denotes the ball of radius (||x_{0}||^{2} + 2cT)^{\frac{1}{2}} e^{cT}. Applying Ascoli’s theorem, we see that the family of all trajectories x of the control system (1) is equicontinuous and bounded, and hence relatively compact in the uniform topology (compare Ekeland [5]).

In order to formulate the multicriteria control problem we introduce the objective function f : R^{n} ⟶ R^{m} and suppose that

(C4) f is a differentiable vector-valued function, (C5) K ⊂ R^{m} is a convex cone with nonempty interior, B ⊂ R^{m} is a pointed convex

cone with int B ≠ ∅ such that cl B + (K \ {0}) ⊂ int B.

Now we formulate the multiobjective optimal control problem under the assumptions (C1) − (C5)

(P): Find some measurable control ū such that the corresponding trajectory x̄ satisfies

f(x(T)) ∉ f(x̄(T)) − (K \ {0})

for all solutions x of (1).

It is well known that it is difficult to show the existence of optimal (or efficient) controls of (P), whereas suboptimal controls exist under very weak conditions. So it is important to derive some assertions for suboptimal controls. An application of a variational principle for vector optimization problems yields an ϵminimum principle for (P), which is closely related to Pontryagin’s minimum principle (for ϵ = 0). Now we apply Theorem 2 in order to derive an ϵ - minimum principle for the multiobjective optimal control problem (P). We introduce the space V of controls, defined as the set of all measurable functions u : [0, T] ⟶ U with the metric

d(u_{1}, u_{2}) = meas{t ∈ [0, T] : u_{1}(t) ≠ u_{2}(t)}.

In order to prove our main result we need the following lemmata:

Lemma 1 (Ekeland [5]) (V, d) is a complete metric space.

Lemma 2 (Ekeland [5]) The function F : u ⟶ f(x(T)) is continuous on V, where x(.) is the solution of (1) depending on u ∈ V.

Theorem 3 Consider the multicriteria control problem under the assumptions (C1)− (C5). For every ϵ > 0, there exists a measurable control u_{ϵ} with the corresponding admissible trajectory x_{ϵ} such that

1. f(x(T)) ∉ f(x_{ϵ}(T)) − ϵk^{0} − (K \ {0})

for all solutions x of (1), 2.

( \begin{matrix}(φ(t, x_{ϵ}(t), u(t)) , p_{ϵ}^{1}(t)) \\ ⋯ \\ (φ(t, x_{ϵ}(t), u(t)) , p_{ϵ}^{m}(t)) \end{matrix} ) ∉ ( \begin{matrix}(φ(t, x_{ϵ}(t), u_{ϵ}(t)) , p_{ϵ}^{1}(t)) \\ ⋯ \\ (φ(t, x_{ϵ}(t), u_{ϵ}(t)) , p_{ϵ}^{m}(t)) \end{matrix} ) − ϵk^{0} − int K

for any u ∈ U and almost all t ∈ [0, T], where p_{ϵ}(.) = (p_{ϵ}^{1}(.), ..., p_{ϵ}^{m}(.)) is the solution of the linear differential system:

\begin{matrix}\frac{dp_{ϵi}^{s}}{dt} (t) = & − ∑_{j=1}^{n} \frac{∂φ_{j}}{∂x_{i}} (t, x_{ϵ}(t), u_{ϵ}(t))p_{ϵj}^{s}, \\ i = & 1, ..., n; \\ p_{ϵ}^{s} = & f′_{s}(x_{ϵ}(T)), \\ s = & 1, ..., m. \end{matrix} \}  (2)

Proof: We consider the vector-valued function F : u ⟶ f(x(T)) and the space V of measurable controls u : [0, T] ⟶ U. From Lemma 1 we get that (V, d) is a complete metric space. Lemma 2 and Remark 3 yield that F : u ⟶ f(x(T)) is lower semicontinuous with respect to k^{0} and B and bounded from below on V. So we can conclude that the assumptions of Theorem 2 are fulfilled and we can apply Theorem 2 to the vector-valued function F. This yields a measurable control u_{ϵ} ∈ V such that

(i) F (u_{ϵ}) ∈ Eff(F [V], K_{ϵk^{0}}), (ii) F (u_{ϵ}) ∈ Eff(F_{ϵk^{0}}[V], K) ,

where

F_{ϵk^{0}}(u) := F (u) + ϵk^{0}d(u, u_{ϵ}).

The corresponding admissible trajectory x_{ϵ} to u_{ϵ} satisfies

\frac{dx_{ϵ}(t)}{dt} = φ(t, x_{ϵ}(t), u_{ϵ}(t))  (3)

for almost all t ∈ [0, T] and

x_{ϵ}(0) = x_{0}.

So we derive from (i)

f(x(T)) ∉ f(x_{ϵ}(T)) − ϵk^{0} − (K \ {0})

for all solutions x of (1), i.e., statement 1 is fulfilled. In order to prove statement 2 we take t_{0} ∈ (0, T), where equality (3) holds, u_{0} ∈ U, and define v_{τ} ∈ V for τ ≥ 0 by: :

v_{τ}(t) := \{ \begin{matrix}u_{0} & : & t ∈ [0, T] ∩ (t_{0} − τ, t_{0}) \\ u_{ϵ}(t) & : & t ∉ [0, T] ∩ (t_{0} − τ, t_{0}) \end{matrix}

For a suffiently small τ it holds:

d(u_{ϵ}, v_{τ}) = meas{t : v_{τ}(t) ≠ u_{ϵ}(t)} ≤ τ.

Furthermore, for the corresponding admissible trajectory x_{τ} it holds: (compare Pallu de la Barriere [18])

\frac{d}{dτ} f_{s}(x_{τ}(T))_{|τ=0} = (φ(t_{0}, x_{ϵ}(t_{0}), u_{0}) − φ(t_{0}, x_{ϵ}(t_{0}), u_{ϵ}(t_{0})) , p_{ϵ}^{s}(t_{0})),  (4)
s = 1, ..., m,

where p_{ϵ} = (p_{ϵ}^{1}, ..., p_{ϵ}^{m}) satisfies (2).

Now, we can conclude from (ii) for u = v_{τ}

F (u_{τ}) + ϵk^{0}d(u_{τ}, u_{ϵ}) ∉ F (u_{ϵ}) − (K \ {0})

and

f(x_{τ}(T)) ∉ f(x_{ϵ}(T)) − ϵk^{0}d(u_{τ}, u_{ϵ}) − (K \ {0}).

For suffiently small τ we get

f(x_{τ}(T)) ∉ f(x_{ϵ}(T)) − ϵτk^{0} − (K \ {0}), τ > 0.

This implies

\frac{f(x_{τ}(T)) − f(x_{ϵ}(T))}{τ} ∉ −ϵk^{0} − (K \ \{0\}) τ > 0

and

lim_{τ→+0} \frac{f(x_{τ}(T)) − f(x_{ϵ}(T))}{τ} ∉ −ϵk^{0} − (K \ \{0\}) τ > 0,

i.e.,

\frac{d}{dτ} f(x_{τ}(T))_{|τ=0} ∉ −ϵk^{0} − int K.

Finally, (4) yields

( \begin{matrix}(φ(t_{0}, x_{ϵ}(t_{0}), u_{0}) − (φ(t_{0}, x_{ϵ}(t_{0}), u_{ϵ}(t_{0})) , p_{ϵ}^{1}(t_{0})) \\ ⋯ \\ (φ(t_{0}, x_{ϵ}(t_{0}), u_{0}) − (φ(t_{0}, x_{ϵ}(t_{0}), u_{ϵ}(t_{0})) , p_{ϵ}^{m}(t_{0})) \end{matrix} ) ∉ −ϵk^{0} − int K

for an arbitrary u_{0} ∈ U and almost all t_{0} ∈ [0, T]. Hence,

( \begin{matrix}(φ(t, x_{ϵ}(t), u(t)) , p_{ϵ}^{1}(t)) \\ ⋯ \\ (φ(t, x_{ϵ}(t), u(t)) , p_{ϵ}^{m}(t)) \end{matrix} ) ∉ ( \begin{matrix}(φ(t, x_{ϵ}(t), u_{ϵ}(t)) , p_{ϵ}^{1}(t)) \\ ⋯ \\ (φ(t, x_{ϵ}(t), u_{ϵ}(t)) , p_{ϵ}^{m}(t)) \end{matrix} ) − ϵk^{0} − int K

for an arbitrary u ∈ U and almost all t ∈ [0, T].

Remark 4: If we put ϵ = 0 then Theorem 3 coincides with the following assertion: Whenever there is a measurable control u_{ϵ} and the corresponding admissible trajectory x_{ϵ} with

1. f(x(T)) ∉ f(x_{ϵ}(T)) − (K \ {0})

for all solutions x of (1),

then 2.

( \begin{matrix}(φ(t, x_{ϵ}(t), u(t)) , p_{ϵ}^{1}(t)) \\ ⋯ \\ (φ(t, x_{ϵ}(t), u(t)) , p_{ϵ}^{m}(t)) \end{matrix} ) ∉ ( \begin{matrix}(φ(t, x_{ϵ}(t), u_{ϵ}(t)) , p_{ϵ}^{1}(t)) \\ ⋯ \\ (φ(t, x_{ϵ}(t), u_{ϵ}(t)) , p_{ϵ}^{m}(t)) \end{matrix} ) − int K

for any u ∈ U and almost all t ∈ [0, T], where p_{ϵ}(.) is the solution of the linear differential system (2).

This means that u_{ϵ} fulfills a minimum principle for multicriteria control problems in the sense of Pontrjagin (compare Gorochowik and Kirillowa [9] and Gorochowik [8], [10]).

Remark 5: Now let us study the special case Y = R^{1} and put ϵ = 0. Then Theorem 3 coincides with the following assertion: Whenever

(i) f(x_{ϵ}(T)) ≤ inf f(x(T)) holds,

then

(ii) (φ(t, x_{ϵ}(t), u_{ϵ}(t)) , p_{ϵ}(t)) ≤ min_{u∈U}(φ(t, x_{ϵ}(t), u(t)) , p_{ϵ}(t))

almost everywhere on [0,T], where p_{ϵ}(.) is the solution of (2).

This is the statement of Pontrjagin’s minimum principle. However, Theorem 3 holds even if optimal solutions do not exist.

In Theorem 3 we have shown an ϵ- minimum principle without any scalarization of the multiobjective optimal control problem. Finally, we derive an ϵ- minimum principle using a suitable scalarization with a functional z : R^{m} ⟶ R^{1} defined for y ∈ R^{m} by:

z(y) := inf{t ∈ R | y ∈ − cl K + tk^{0}}.  (5)

Definition 2 Let D be a nonempty subset of R^{m} and y^{1}, y^{2} ∈ R^{m}. A functional z : R^{m} ⟶ R is called D-monotone, if y^{1} ∈ y^{2} + D implies z(y^{1}) ≥ z(y^{2}). A Dmonotone functional z is called strictly D-monotone, if y^{1} ∈ y^{2} + (D \ {0}) implies z(y^{1}) > z(y^{2}).

In [7], Corollary 2.1, under the assumption that K is a convex cone with nonempty interior, we have shown that the functional z in (5) is continuous, sublinear and strictly (int K) - monotone.

Theorem 4 Consider the multicriteria control problem under the assumptions (C1)− (C5). For every ϵ > 0, there exists a measurable control u_{ϵ} with the corresponding admissible trajectory x_{ϵ} such that

1. f(x(T)) ∉ f(x_{ϵ}(T)) − ϵk^{0} − (K \ {0})

for all solutions x of (1), 2.

z(( \begin{matrix}(φ(t, x_{ϵ}(t), u(t)) − φ(t, x_{ϵ}(t), u_{ϵ}(t)) , p_{ϵ}^{1}(t)) \\ ⋯ \\ (φ(t, x_{ϵ}(t), u(t)) − φ(t, x_{ϵ}(t), u_{ϵ}(t)) , p_{ϵ}^{m}(t)) \end{matrix} )) ≥ −ϵ,

for any u ∈ U and almost all t ∈ [0, T], where z : R^{m} ⟶ R^{1} is the continuous, sublinear, strictly (int K) - monotone functional defined by (5) and p_{ϵ} = (p_{ϵ}^{1}, ..., p_{ϵ}^{m}) is the solution of the linear differential system (2).

Proof: Under the given assumptions we can conclude from the statement 2 of Theorem 3

( \begin{matrix}(φ(t, x_{ϵ}(t), u(t)) − φ(t, x_{ϵ}(t), u_{ϵ}(t)) , p_{ϵ}^{1}(t)) \\ ⋯ \\ (φ(t, x_{ϵ}(t), u(t)) − φ(t, x_{ϵ}(t), u_{ϵ}(t)) , p_{ϵ}^{m}(t)) \end{matrix} ) ∉ −ϵk^{0} − int K  (6)

for any u ∈ U and almost all t ∈ [0, T], where p_{ϵ} = (p_{ϵ}^{1}, ..., p_{ϵ}^{m}) is the solution of (2). The functional z in (5) has for y ∈ R^{m} and t ∈ R the property (cf. [7], Theorem 2.1)

z(y) ≥ t ⟺ y ∉ − int K + tk^{0},

such that we can conclude from (6)

z(( \begin{matrix}(φ(t, x_{ϵ}(t), u(t)) − φ(t, x_{ϵ}(t), u_{ϵ}(t)) , p_{ϵ}^{1}(t)) \\ ⋯ \\ (φ(t, x_{ϵ}(t), u(t)) − φ(t, x_{ϵ}(t), u_{ϵ}(t)) , p_{ϵ}^{m}(t)) \end{matrix} )) ≥ −ϵ,

for any u ∈ U and almost all t ∈ [0, T], where p_{ϵ}(.) is the solution of (2).

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