# Nonessential functionals in multiobjective optimal control Problems/mitteolulised funktsionaalid mitmekriteeriumilistes optimaaljuhtimise ulesannetes.

1. INTRODUCTIONMultiobjective optimal control attracts more and more attention and is a topic of extensive current research (see, e.g., [1-3] and references therein). We consider multiobjective problems of optimal control governed by ordinary differential dynamical systems. This comprises an important class of problems which naturally appear in practical applications to economics [4], Ch. 8, and engineering modelling [5]. Our main goal is to extend and apply the results found in the literature on nonessential functions of mathematical static optimization programming [6,7] to functionals of optimal control theory.

It is well known that the concept of Pareto optimality (efficiency) plays a crucial role in optimal control [5,8]. The question of obtaining well-defined criteria for multiple criteria decision-making problems seems, however, to be considered in the literature only for static multiobjective optimization problems (cf. [6,7] and references therein). In this work we investigate the problem of obtaining well-defined criteria for multiple criteria optimal control problems.

One of the approaches dealing with the problem of obtaining well-defined criteria for multiple criteria static decision-making problems is the concept of nonessential objective functions. A certain objective function is called nonessential if the set of efficient solutions is the same both with or without that objective function. Information about nonessential objectives helps a decision-maker to get insights and understand better a problem, and this might be a good starting point for further investigation or revision of the model. Dropping nonessential functions leads to a problem with a smaller number of objectives, which can be solved more easily. For this reason, the idea of nonessential objectives should be used as a regular feature in conditioning and analysis of multiple criteria programs [6,9,10]. To the best of the authors' knowledge, no study has been done in this field for optimal control problems. We are interested in generalizing the previous results on nonessential objectives found in the literature to cover multiobjective optimal control problems. More precisely, we generalize the concept of nonessential objective functions to functionals of optimal control systems and give the first steps on the corresponding theory. The main results provide methods for identifying nonessential objectives in multiobjective optimal control problems.

2. MULTIOBJECTIVE OPTIMAL CONTROL PROBLEMS

We consider a dynamical control system described by n state variables x = ([x.sub.1],..., [x.sub.n]) [member of] [R.sup.n] and r control variables u = ([u.sub.1],..., [u.sub.r]) [member of] [R.sup.r], r [less than or equal to] n. Both state and control variables vary with respect to the scalar variable t [member of] R. Given a control vector function u : [a, b] [right arrow] [R.sup.r], the state evolution over [a, b], namely x : [a, b] [right arrow] [R.sup.n], must satisfy the control system

[??](t) = h(t, x(t), u(t)), (1)

the boundary conditions

x(a) = [alpha], x(b) = [beta], (2)

and m inequality constraints

[g.sub.i](t, x(t), u(t)) [less than or equal to] 0, i = 1,...m. (3)

We would like to find a piecewise-continuous control function u(*) and the corresponding state trajectory x(*), satisfying (1), (2), and (3), which minimizes a finite number N of (objective) functionals:

min [[integral].sup.b.sub.a] f(t, x(t), u(t))dt

= min ([[integral].sup.b.sub.a] [f.sub.1](t, x(t), u(t))dt, ... , [[integral].sup.b.sub.a] [f.sub.N](t, x(t), u(t))dt).

All functions f(t, x, u), g(t, x, u), and h(t, x, u) are assumed to be continuously differentiable with respect to t and x variables. To simplify notation, we write

[I.sup.N][x, u] = [[integral].sup.b.sub.a] f(t, x(t), u(t))dt

and

[I.sub.i][x, u] = [[integral].sup.b.sub.a] [f.sub.i](t, x(t), u(t))dt, i = 1,...,N.

In general, there does not exist a pair of functions (x, u) that renders the minimum value to each functional [I.sub.i], i = 1,...,N, simultaneously, and one uses the concept of Pareto optimality. Let us denote by S the set of feasible solutions, i.e. the set of all admissible functions (x, u). The multiobjective optimal control problem consists of finding all feasible solutions that are efficient in the sense of Definition 2.1. This problem is denoted in the sequel by (P).

Definition 2.1. (Pareto optimality). A pair of functions ([??], [??]) [member of] S is said to be an efficient (Pareto optimal) solution of the problem (P) if, and only if, there exists no (x, u) [member of] S such that [I.sup.N][x, u] [??] [I.sup.N][[??], [??]], where

[I.sup.N][x, u] [??] [I.sup.N][[??], [??]] [??] [for all]i [member of] {1,...,N} : [I.sub.i][x, u] [less than or equal to] [I.sub.i][[??], [??]][conjunction] [there exists]j [member of]{1,...,N} : [I.sub.j] [x, u] < [I.sub.j] [[??], [??]].

The set of efficient solutions of (P) is denoted by [S.sup.N.sub.E].

We remark that many practical applications that appear in engineering and economics can be written in the form of problem (P) [5].

The central result in optimal control theory is given by the celebrated Pontryagin maximum principle [11], which is a necessary optimality condition. A version of the Pontryagin maximum principle for Pareto optimal solutions of multiobjective optimal control problems was proved already in the 1960s [12]. Roughly speaking, one can say that the necessary and sufficient conditions for Pareto optimality are obtained by converting the multiobjective optimal control problem into a single optimal control problem or a family of single optimal control problems with an auxiliary scalar integral functional, possibly depending on some parameters [13,14]. For a gentle introduction to optimal control, including necessary and sufficient conditions and the issue of existence, we refer the reader to [15,16] (scalar case) and [5,8] (Pareto optimal solutions). Here we just recall three basic lemmas (cf. [8], Ch. 17) that relate the Pareto optimal solution of a multiobjective optimal control problem to the solutions of an appropriate scalar optimal control problem.

Lemma 2.2. ([8], [section]17.4) If the feasible pair ([??], [??]) [member of] S is efficient for (P), then it minimizes each one of the scalar integral functionals

[I.sub.i][x, u], i [member of] {1,...,N},

subject to the constraints (x, u) [member of] S and

[I.sub.j] [x, u] - [I.sub.j] [[??], [??]] [less than or equal to] 0, j = 1,...,N (and j [not equal to] i).

Lemma 2.2 is very useful because it implies that the necessary conditions for optimal control subject to isoperimetric constraints [11,17] are also necessary for Pareto optimality in the multiobjective optimal control problem. As with the necessary conditions, the next two lemmas reduce the sufficient conditions for Pareto optimality to sufficient conditions for optimal control in the scalar optimal control problem.

Lemma 2.3. ([8], [section]17.5) A feasible pair ([??], [??]) [member of] S is efficient for (P) if there exists a constant [gamma] [member of] [R.sup.N], with [[gamma].sub.i] > 0 for i = 1,...,N and [[summation].sup.N.sub.i=1] [[gamma].sub.i] = 1, such that

[N.summation over (i=1)] [[gamma].sub.i][I.sub.i][x, u] [greater than or equal to] [N.summation over (i=1)] [[gamma].sub.i][I.sub.i][[??], [??]]

for every (x, u) [member of] S.

Remark 2.4. Proof of Lemma 2.3 is very simple. Moreover, Lemma 2.3 is easily generalized. Let f be a strongly increasing function if y [??] [??] [??] f(y) < f([??]). Then ([??], [??]) [member of] S is efficient for (P) if there exists a strongly increasing function f and f ([I.sup.N][x, u]) [greater than or equal to] f ([I.sup.N][[??], [??]])[for all](x, u) [member of] S.

Unlike Lemma 2.3, not all the components of [gamma] in the next Lemma 2.5 need to be nonzero. However, in Lemma 2.5 the minimum of [[summation].sup.N.sub.i=1] [[gamma].sub.i][I.sub.i][x, u] must be achieved by a unique ([??], [??]) [member of] S.

Lemma 2.5. ([8], [section]17.5) A feasible pair ([??], [??]) [member of] S is efficient for (P) if there exists a constant [gamma] [member of] [R.sup.N], with [[gamma].sub.i] [greater than or equal to] 0 for i = 1,...,N and [[summation].sup.N.sub.i=1] [[gamma].sub.i] = 1, such that

[N.summation over (i=1)] [[gamma].sub.i][I.sub.i][x, u] > [N.summation over (i=1)] [[gamma].sub.i][I.sub.i][[??], [??]]

for every (x, u) [member of] S, (x, u) [not equal to] ([??], [??]).

Together with the Pontryagin maximum principle [11,17], Lemmas 2.2, 2.3, and 2.5 provide expedient tools to study concrete multiobjective problems of optimal control (cf. Section 4).

3. NONESSENTIAL FUNCTIONALS: MAIN RESULTS

We form a new multiobjective optimal control problem ([??]) from (P) by adding a new functional [I.sub.N+1][x, u] = [[integral].sup.b.sub.a][f.sub.N+1](t, x(t), u(t))dt to problem (P). Let [S.sup.N+1.sub.E] denote the set of efficient solutions of the problem ([??]). With this notation we introduce the definition of a nonessential functional.

Definition 3.1. The functional [I.sub.N+1] is said to be nonessential in ([??]) if, and only if, [S.sup.N.sub.E] = [S.sup.N+1.sub.E]. A functional which is not nonessential will be called essential.

We are interested in characterizing integral functionals which do not change the set of efficient solutions (nonessential objective functionals). Throughout the text we denote by [S.sub.i], i = 1, 2,...,N,N + 1, the set of optimal solutions of the scalar optimal control problem

min [I.sub.i][x, u]

subject to S. We start with a simple example taken from [18].

Example 3.2. Consider a system characterized by a single state and control variable (n = r = 1) that evolves according to the state equation

[??](t) = u(t)

with the control constraint set

U = {u : [a, b] [right arrow] R : |u(t)| [less than or equal to] 1}.

The system is to be transferred from a given initial state x(0) = [xi] [not equal to] 0 to a given terminal state x(T) = 0 within an unspecified bounded interval [0, T]. Functionals to be minimized are

[I.sub.1] = [[integral].sup.T.sub.0] dt, [I.sub.2] = [[integral].sup.T.sub.0] |u(t)|dt.

Applying the Pontryagin maximum principle [11], one obtains:

[S.sub.1] = {(x(t), u(t)) : u(t) = -sgn{[xi]}, min [[integral].sup.T.sub.0] dt = |[xi]|

and

[S.sub.2] = {(x(t), u(t)) : u(t) = -sgn{[xi]}v(t)},

where

v(t) [member of] V = {v(t) : 0 [less than or equal to] v(t) [less than or equal to] 1, t [member of] [0, T], v(t) [not equal to] 0}, min [[integral].sup.T.sub.0] | -sgn{[xi]}v(t)|dt = |[xi]|.

Details can be found in [18]. It is easy to see that [S.sub.1] [intersection] [S.sub.2] = [S.sub.1] (we can take v(t) = 1, t [member of] [0, T]). In this problem we have: [S.sub.1] = [S.sup.1.sub.E] = [S.sup.2.sub.E] [subset] [S.sub.2]. Hence [I.sub.2] is nonessential, but [I.sub.1] is essential (in order to see this, we need only to interchange indices).

Lemma 3.3. We have [S.sup.N.sub.E] [subset] [S.sup.N+1.sub.E] if and only if for every (x, u) [member of] [S.sup.N.sub.E] the following condition holds:

[there exists](x', u') [member of] S : [I.sup.N][x', u'] = [I.sup.N][x, u] [??] [I.sub.N+1][x', u'] = [I.sub.N+1][x, u].

Proof. Let [S.sup.N.sub.E] [subset] [S.sup.N+1.sub.E] and assume, on the contrary, that there exists ([??], [??]) [member of] [S.sup.N.sub.E] such that

9(x', u') [member of] S : [I.sup.N][x', u'] = IN[[??], [??]] (4)

and

[I.sub.N+1][x', u'] [not equal to] [I.sub.N+1][[??], [??]]. (5)

We conclude from (4) that (x', u') [member of] [S.sup.N.sub.E]. Therefore (x', u') is not in [S.sup.N+1.sub.E] or ([??], [??]) is not in [S.sup.N+1.sub.E] by (5). This contradicts the fact that [S.sup.N.sub.E] [subset] [S.sup.N+1.sub.E] . Let us now prove the second implication. If [S.sup.N.sub.E] = [empty set], then [S.sup.N.sub.E] [subset] [S.sup.N+1.sub.E]. Let [S.sup.N.sub.E] [not equal to] [empty set]. Suppose that for every (x, u) [member of] [S.sup.N.sub.E] there holds:

[there exists](x', u') [member of] S : [I.sup.N][x', u'] = [I.sup.N][x, u] [??] [I.sub.N+1][x', u'] = [I.sub.N+1][x, u] (6)

and [S.sup.N.sub.E] is not contained in [S.sup.N+1.sub.E] . In this case there exists ([??], [??]) in [S.sup.N.sub.E] which is not in [S.sup.N+1.sub.E] . Hence

[there exists]([??], [??]) [member of] S : [I.sup.N+1][[??], [??]] [??] [I.sup.N+1][[??], [??]]. (7)

This gives IN[[??], [??]] = [I.sup.N][[??], [??]] and from (6) we have [I.sub.N+1][[??], [??]] = [I.sub.N+1][[??], [??]]. Consequently, [I.sup.N+1][[??], [??]] = [I.sup.N+1][[??], [??]], contrary to (7).

Remark 3.4. Notice that in Example 3.2 the scalar optimal control problem

min [I.sub.1][x, u]

subject to S has a unique solution. Therefore, Lemma 3.3 holds true for the example.

Definition 3.5. A function f : [R.sup.N] [right arrow] R is nondecreasing if for [y.sup.1] and [y.sup.2] [member of] [R.sup.N] [y.sup.1] [??] [y.sup.2] implies f([y.sup.1]) [less than or equal to] f([y.sup.2]).

Theorem 3.6. If [I.sub.N+1][x, u] = f([I.sup.N][x, u]) [for all] (x, u) [member of] S, where f : [R.sup.N] [right arrow] R, then [S.sup.N.sub.E] [subset] [S.sup.N+1.sub.E] . Furthermore, [S.sup.N.sub.E] = [S.sup.N+1.sub.E] if the function f is nondecreasing on the set [I.sup.N](S).

Proof. Let (x, u) [member of] [S.sup.N.sub.E]. If there exists (x', u') [member of] S such that [I.sup.N][x', u'] = [I.sup.N][x, u], then f([I.sup.N][x', u']) = f([I.sup.N][x, u]) and so [I.sub.N+1][x', u'] = [I.sub.N+1][x, u]. Therefore [S.sup.N.sub.E] [subset] [S.sup.N+1.sub.E] by Lemma 3.3.

We will now show the inclusion [S.sup.N+1.sub.E] [subset] [S.sup.N.sub.E]. Let (x, u) [member of] S, (x, u) being not an element of the set [S.sup.N.sub.E]. In this case there exists (x', u') [member of] S such that [I.sup.N][x', u'] [??] [I.sup.N][x, u]. If f is nondecreasing on [I.sup.N](S), we know that [I.sub.N+1][x', u'] = f([I.sup.N][x', u']) [less than or equal to] f([I.sup.N][x, u]) = [I.sub.N+1][x, u]. Hence (x, u) is not an element of the set [S.sup.N+1.sub.E] .

Remark 3.7. Example 3.2 shows that the sufficient condition for an objective functional to be nonessential, given by Theorem 3.6, is not necessary.

Theorem 3.8. Let [S.sub.N+1] = {(x', u')}. If the functional [I.sub.N+1] is nonessential, then (x', u') [member of] [S.sup.N.sub.E].

Proof. Let [S.sup.N.sub.E] = [S.sup.N+1.sub.E] . If (x', u') is not an element of the set [S.sup.N.sub.E], then (x', u') is also not an element of the set [S.sup.N+1.sub.E] . In this case, there exists (x', u') [member of] S such that [I.sup.N+1][x', u'] [??] [I.sup.N+1][x', u']. So [I.sub.N+1][x', u'] [less than or equal to] [I.sub.N+1][x', u']. This is a contradiction to the assumption that [S.sub.N+1] = {(x', u')}.

Theorem 3.9. Let the set S be compact. If the functional [I.sub.N+1] is nonessential, then [S.sub.N+1] [intersection] [S.sup.N.sub.E] [not equal to] [empty set].

Proof. Consider the problem

min [[integra].sup.b.sub.a] f(t, x(t), u(t))dt (8)

subject to [S.sub.N+1]. Let [??] denote the set of efficient solutions of the problem (8). By the compactness of the set S, the set [??] is nonempty. Let (x', u') [member of] [??]. If (x', u') is not an element of [S.sup.N.sub.E] , then by assumption (x', u') is not an element of [S.sup.N+1.sub.E]. In this case, there exists (x', u') [member of] S such that [I.sup.N+1][x', u'] [??] [I.sup.N+1][x', u']. Hence (x', u') [member of] [S.sub.N+1]. This contradicts the fact that (x', u') is an efficient solution of the problem (8).

Remark 3.10. Notice that [I.sub.2] is nonessential in Example 3.2 and we have [S.sup.1.sub.E] [intersection] [S.sub.2] [not equal to] [empty set].

The next section provides an example of application of the obtained results to check whether a functional is nonessential.

4. AN ILLUSTRATIVE EXAMPLE

We illustrate the obtained results with a multiobjective control problem borrowed from [5], [section]4.3, where N = 3, n = 2, r = 1, m = 4, a = 0, b = T, with T not fixed. We consider a mobile rocket car with mass one running in rails on a closed region -3 [less than or equal to] [x.sub.1] [less than or equal to] 3 (we denote the position of the centre of the car at time t by [x.sub.1](t)), whose movement we can control with its accelerator u, where the maximum allowable acceleration is 1 and the maximum break power is -1, i.e., -1 [less than or equal to] u [less than or equal to] 1 (negative force means break, positive force means acceleration). The dynamics of the system is given by Newton's second law, force equals mass times acceleration, which in our setting reads as u(t) = [[??].sub.1](t). The problem is to move the car from a given location to a pre-assigned destination. If the car is at a position [x.sub.1] = 1 at time t = 0, with no velocity, that is [[??].sub.1](0) = 0, we want to find a piecewise constant function u(t) that drives the car to [x.sub.1](T) = 0 at some instant T > 0. The state of the system is given by the position [x.sub.1](t) and the velocity [x.sub.2](t) = [[??].sub.1](t) (where we are and how fast we are going at each instant of time t). Different objective functionals can be considered, for example, minimizing the time T (functional [I.sub.1] below); maximizing the velocity at T (maximizing [x.sub.2](T), which corresponds to functional [I.sub.2] below); and a linear combination [I.sub.3] of these functionals: minimize

[I.sub.1] = [[integral].sup.T.sub.0] 1dt, [I.sub.2] = [[integral].sup.T.sub.0] - u(t)dt, [I.sub.3] = [I.sub.1] + [I.sub.2],

subject to the control system

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (9)

to the boundary conditions

[x.sub.1](0) = 1, [x.sub.1](T) = 0, [x.sub.2](0) = 0, (10)

and inequality constraints

|u| [less than or equal to] 1, |[x.sub.1]| [less than or equal to] 3. (11)

Let us denote by [S.sub.i], i = 1, 2, 3, the solution set of the scalar optimal control problem min [I.sub.i][x, u] subject to (9)-(11). We have: (1) [S.sub.1] = {([x.sub.1], [u.sup.1])} with

[u.sup.1] = -1, [x.sup.1.sub.1] = - [t.sup.2]/2 + 1, [x.su[.1.sub.2] = -t, 0 [less than or equal to] t [less than or equal to] T = [square root of 2],

[S.sub.2] = {([x.sup.2], [u.sup.2])} with

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

[S.sub.3] = {([x.sup.3], [u.sup.3])} with

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Direct calculations show that

[I.sup.2][[x.sub.1], [u.sup.1]] = [[[square root of 2], [square root of 2]].sup.T] = A, [I.sup.2][[x.sup.2], [u.sup.2]] = [[4 + [square root of 6], - [square root of 6]].sup.T] = B, [I.sup.2][[x.sup.3], [u.sup.3]] = [[2, 0].sup.T] = C.

Let [zeta] denote the set [I.sup.2]([S.sup.2.sub.E]). It is the continuous, convex curve [??] (details can be found in [5], [section]4.3). As C [member of] [zeta], we have [S.sup.2.sub.E] [intersection] [S.sub.3] [not equal to] [empty set]. Moreover, let us notice that [I.sub.3] has the form [I.sub.3][x, u] = f([I.sup.2][x, u]), where f : [R.sup.2] [right arrow] R is a nondecreasing function. Therefore, the functional [I.sub.3] is nonessential by Theorem 3.6.

Remark 4.1. If we change the functional [I.sub.3] into

[I.sub.3][x, u] = [[gamma].sub.1][I.sub.1][x, u] + [[gamma].sub.2][I.sub.2][x, u], (12)

where [[gamma].sub.i] [member of] R and [[gamma].sub.i] [greater than or equal to] 0, i = 1, 2 or

[I.sub.3][x, u] = [[[([I.sub.1][x, u] - [square root of 2]).sup.p] + [([I.sub.2][x, u] + [square root of 6]).sup.p]].sup.1/p], (13)

where p [member of] [1,[infinity]], then again [I.sub.3] will be nonessential by Theorem 3.6. It is worth noting that functionals (12) and (13) can be used in order to find efficient solutions of the problem min [I.sup.2][x, u] subject to (9)-(11). We mentioned this in Section 2, details can be found in [5] and [8].

5. CONCLUSIONS

The problem of optimization of a vector-valued criterion often arises in connection with the solution of problems in the areas of planning, organization of production, operational research, and dynamical control systems. Currently, the problem of optimizing vector-valued criteria is a central part of control theory and great attention is being given to it in the design and construction of modern automatic control systems, for example in specific applications of seismology, energetic chemistry, and metallurgy. In this work we use the notion of Pareto optimality in control theory to define and investigate nonessential objective functionals of optimal control. For multiple criteria optimal control problems this notion seems to be new and not used before. We believe that the concept of a nonessential objective functional is an important issue in optimal control and we trust it will have an important role in the study of vector optimization problems of control theory. In future, it would be interesting to study the consequences of dropping nonessential objectives in multiobjective optimal control problems.

ACKNOWLEDGEMENTS

Agnieszka B. Malinowska was supported by KBN under Bialystok Technical University grant No. W/WI/17/07, Delfim F. M. Torres by the R&D unit "Centre for Research in Optimization and Control" (CEOC). The authors are grateful to Ignacy Kaliszewski and Domingos Cardoso for useful comments; to Enrique Hernandez-Manfredini for the suggestions regarding improvement of the text.

Received 15 March 2007, in revised form 4 July 2007

REFERENCES

[1.] Aubin, J.-P. and Frankowska, H. Set-Valued Analysis, Systems & Control: Foundations & Applications, Vol. 2. Birkhauser Boston, Boston, MA, 1990.

[2.] Hamel, A. H. Optimal control with set-valued objective function. In Proceedings of the 6th Portuguese Conference on Automatic Control - Controlo 2004, Faro, Portugal. 2004, 648-652.

[3.] Torres, D. F. M. A Noether theorem on unimprovable conservation laws for vector-valued optimization problems in control theory. Georgian Math. J., 2006, 1, 173-182.

[4.] Mordukhovich, B. S. Variational Analysis and Generalized Differentiation, II. Springer, New York, 2006.

[5.] Salukvadze, M. E. Vector-Valued Optimization Problems in Control Theory. Academic Press, New York, 1979.

[6.] Gal, T. and Hanne, T. Nonessential objectives within network approaches for MCDM. European J. Oper. Res., 2006, 2, 584-592.

[7.] Malinowska, A. B. Nonessential objective functions in linear vector optimization problems. Control Cybern., 2006, 35, 873-880.

[8.] Leitmann, G. The Calculus of Variations and Optimal Control. Plenum, New York, 1981.

[9.] Gal, T. and Leberling, H. Redundant objective functions in linear vector maximum problems and their determination. European J. Oper. Res., 1977, 1, 176-184.

[10.] Malinowska, A. B. Changes of the set of efficient solutions by extending the number of objectives and its evaluations. Control Cybern., 2002, 31, 964-974.

[11.] Pontryagin, L. S. and Boltyanskii, V. G. The Mathematical Theory of Optimal Processes. Interscience Publishers John Wiley & Sons, Inc., New York, 1962.

[12.] Chang, S. S. L. General theory of optimal processes. SIAM J. Control, 1966, 4, 46-55.

[13.] Liu, L. P. Characterization of nondominated controls in terms of solutions of weighting problems. J. Optim. Theory Appl., 1993, 77, 545-561.

[14.] Reid, R. W. and Citron, S. J. On noninferior performance index vectors. J. Optim. Theory Appl., 1971, 7, 11-28.

[15.] Macki, J.W. and Strauss, A. Introduction to Optimal Control Theory. Springer, New York, 1982.

[16.] Pedregal, P. Introduction to Optimization. Springer, New York, 2004.

[17.] Ferreira, M. M. and de Pinho, M. d. R. Optimal Control Problems with Constraints. Editura Electus, Bucharest, 2002.

[18.] Athans, M. and Falb, P.-L. Optimal Control: An Introduction to the Theory and Its Applications. McGraw-Hill, New York, 1966.

(1) The solutions to the scalar optimal control problems [I.sub.i][x, u] [right arrow] min are found by application of the Pontryagin maximum principle [11]. Details can be found in [5], x4.3.

Partially presented at the 5th Junior European Meeting on Control and Information Technology (JEM'06), September 20-22, 2006, Tallinn, Estonia.

Agnieszka B. Malinowska (a) and Delfim F. M. Torres (b)

(a) Faculty of Computer Science, Bialystok Technical University, Wiejska 45A, 15-351 Bialystok, Poland; abmalina@pb.edu.pl

(b) Department of Mathematics, University of Aveiro, 3810-193 Aveiro, Portugal; delfim@ua.pt