The first study of differential equations with multivalued right-hand sides was performed by A. Marchaud [1,2] and S.C. Zaremba [3]. In early sixties, T. Wazewski [4,5], A.F. Filippov [6] obtained fundamental results on existence and properties of the differential equations with multivalued right-hand sides (differential inclusions). One of the most important results of these articles was an establishment of the relation between differential inclusions and optimal control problems, that promoted to develop the differential inclusion theory [7].

Considering of the differential inclusions required to study properties of multivalued functions, i.e. an elaboration the whole tool of mathematical analysis for multivalued functions [8-10].

Simultaneously there were appeared papers [11-14] which used Hukuhara derivative [9,10] of multivalued function for investigation of differential equations with multivalued right-hand sides and solutions.

In works [15,16] annotate of an R-solution for differential inclusion is introduced as an absolutely continuous multivalued function. Various problems for the R-solution theory were regarded in [17-22].

The basic idea for a development of an equation for R-solutions (integral funnels) is contained in [23].

Here the equation was considered as a particular case of an approximated equation in a metric space. Approximated equations make possible to exclude an differentiation operation and there by to avoid linearity requirement for a solution space and to consider differential equations in linear metric spaces, equations with multivalued solutions and dynamical systems in nonlinear metric spaces by unified positions [14,20,23,24].

Therefore, an approximated equation is a quasidifferential equation for determination of the dynamical system in a metric space.

The theory of mutational equations in metric spaces, which deals with multivalued trajectories (pipers) and trajectories in nonlinear spaces has been developed in [25].

As well as in [23] it is taken an approach that does not use a derivative in explicit form for description in nonlinear metric spaces, while in [25] analogous results are obtained by construction "differential calculus" in nonlinear metric spaces.

Moreover in [23] quasidifferential equations were considered for locally compact spaces, while in [20] for complete metric space .

In the last years there has been forming new approach to control problems of dynamic systems, which foundation on analysis of trajectory bundle but not separate trajectories. The section of this bundle in any instant is some set and it is necessary to describe the evolution of this set. Obtaining and research dynamic equations of sets there is important problem in this case. The metric space of sets with the Hausdorff metric is natural space for description dynamic of sets. In theory of multivalued maps definitions on derivative as for single-valued maps is impossible because space of sets is nonlinear. This bound possibility description dynamic sets by differential equations. Therefore, the control differential equations with set of initial conditions [26-28], the control differential inclusions [29-40], the control differential equations with Hukuhara derivative [14] and the control quasidifferential equations [14,40,41] use for it.

In recent years, the fuzzy set theory introduced by Zadeh [42] has emerged as an interesting and fascinating branch of pure and applied sciences. The applications of fuzzy set theory can be found in many branches of regional, physical, mathematical, differential equations, and engineering sciences. Recently there have been new advances in the theory of fuzzy differential equations [43-55] and inclusions [50,56-59] as well as in the theory of control fuzzy differential equations [60-62] and inclusions [63-65].

In this article we consider the some properties of the fuzzy R-solution of the control linear fuzzy differential inclusions and research the optimal time problems for it.

2. The Control Differential inclusions

2.1. The Fundamental Definitions and Designations

Let comp([R.sup.n])(conv([R.sup.n])) be a set of all nonempty (convex) compact subsets from the space [R.sup.n]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

be Hausdorff distance between sets A and B, [S.sub.r](A) is r-neighborhood of set A.

Let [E.sup.n] be the set of all u:[R.sup.n] [right arrow][0,1] such that u satisfies the following conditions:

1) u is normal, that is, there exists an [x.sub.0][member of][R.sup.n] such that u([x.sub.0)=1;

2) u is fuzzy convex, that is, u([lambda]x + (1-[lambda])y)[greater than or equal to] min{u(x),u(y)}

3) for any x, y [member of] [R.sup.n] and 0[less than of equal to][lambda][less than or equal to]1;

4) u is upper semicontinuous,

5) [[u].sup.0] = cl[x [member of] [R.sup.n]:u(x)>0} is compact.

If u [member of] [E.sup.n], then u is called a fuzzy number, and [E.sup.n] is said to be a fuzzy number space. For 0 < [alpha] [less than or equal to] 1, denote [[u].sup.[alpha]] = {x [member of] [R.sup.n]:u(x) [greater than or equal to] [alpha]}.

Then from 1)-4), it follows that the [alpha]-level set [[u].sup.[alpha]] [member of] conv([R.sup.n]) for all 0 [less than or equal to] [alpha] [less than or equal to] 1.

Theorem 1 (Negoita and Ralescu [66]). If , u [member of] [E.sup.n], then

1) [[u].sup.[alpha] [member of] conv ([R.sup.n]) for all [alpha] [member of] [0,1];

2) [[u].sup.[alpha] [subset] [[u].sup.[beta]] for 0 [less than or equal to] [alpha] < [beta] [less than or equal to] 1;

3) If {[[alpha].sub.k]} [subset] [0,1] is a decreasing sequence converging to [alpha] > 0 then [[u].sup.[alpha]]= [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Conversely, if {[A.sup.[alpha]]: 0 [less than or equal to] [alpha] [less than or equal to] 1}is a family of convex compact subsets of [R.sup.n] satisfying 1)-3), then [[u].sup.[alpha]] = [A.sup.[alpha] for 0 < [alpha] [less than or equal to] 1 and [[u].sup.0] = [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

If g: [Rsup.n] x [Rsup.n] [right arrow] [Rsup.n] is a function, then using Zadeh's extension principle we can extend [??] to [E.sup.n] x [E.sup.n] [right arrow] [E.sup.n] by the equation

[??](u,v)(z) = sup min[u(x),v(y)]. z=g(x,y)

It is well known that

[[[??](u,v].sup.[alpha]] = g([[u].sup.[alpha]], [[v].sup.[alpha]]) for all u, v [member of] [E.sup.n], 0 [less than or equal to] [alpha] [less than or equal to] 1 and continuous function g.

Further, we have

[[u+v].sup.[alpha]]=[[u].sup.[alpha]]+ [[v].sup.[alpha]], [[ku].sup.[alpha]] = k[[u].sup.[alpha]], where k [member of] R.

Define D:[E.sup.n] x [E.sup.n] [right arrow] [0,[infinity]) by the relation

D(u,v0 = [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where h is the Hausdorff metric defined in comp([R.sup.n]).

Then D is a metric in [E.sup.n].

Further we know that [67]:

1) ([E.sup.n],D)is a complete metric space,

2) D(u+w,v+w)=D(u,v) for all u,v,w [member of] [E.sup.n],

3) D([lambda]u, [lambda]v)= |[lambda]]| D(u,v)for all u, v [member of] [E.sup.n] and [lambda] [member of] R.

It can be proved that D(u+v,w+z)[less than or equal to]D(u,w)+D(v,z) for u,v,w,z [member of] [E.sup.n]

Definition 1. A mapping F :[0,T] [right arrow] [E.sup.n] is strongly measurable if for all [alpha] [member of][0,1] the set-valued map [F.sub.[alpha]]:[0,T] [right arrow] conv([R.sup.n])defined by [F.sub.[alpha]](t) = [[F(t)].sup.[alpha]] is Lebesgue measurable.

Definition 2. A mapping F:[0,T] [right arrow] [E.sup.n] is said to be integrably bounded if there is an integrable function h(t) such that [parallel]x(t)[parallel] [less than or equal to] h(t) for every x(t) [member of] [F.sub.0](t).

Definition 3. The integral of a fuzzy mapping F:[0,T] [right arrow] [E.sup.n] is defined levelwise by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. The set of all 0 [integral]T f(t)dt such that

f:[0,T] [right arrow] [R.sup.n] is a measurable selection for [F.sub.[alpha]]for all [alpha] [member of] [0,1].

Definition 4. A strongly measurable and integrably bounded mapping F:[0,T] [right arrow] [E.sup.n] is said to be integrable over [0,T]if [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Note that if F:[0,T] [right arrow] [E.sup.n] is strongly measurable and integrably bounded, then F is integrable. Further if F:[0,T] [right arrow] [E.sup.n] is continuous, then it is integrable.

Theorem 2. [43]. Let F,G:[0,T] [right arrow][E.sup.n] be integrable and c [member of] [0,T], [lambda] [member of] R. Then

a) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII];

b) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII];

c) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII];

d) D(F,G) is integrable; .

e) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Consider the following control linear fuzzy differential inclusions

[??][member of] A(t)x + G(t,w), x([t.sub.0])= [x.sub.0], (1) and the following nonlinear fuzzy differential inclusions

[??] [member of] F(T,x,w), x([t.sub.0])= [x.sub.0],, (2) where [??] means dx/dt; t [member of] [R.sub.+]is the time; x [member of][R.sup.n] is the state; w [member of] [R.sup.m]is the control; A(t) is (nxn)- dimensional matrix-valued function; G:[R.sub.+] x [R.sup.m] [right arrow] [E.sup.n], F: [R.sub.+] x [R.sup.n] x [R.sup.m] [right arrow] [E.sup.n] are the set-valued functions.

Let

W:[R.sub.+] [right arrow]conv([R.sup.m]) (3) be the measurable multivalued map.

Definition 5. Set LW of all single-valued branches of the multivalued map W([??]) is the set of the possible controls.

Obviously, the control fuzzy differential inclusion (2) turns into the ordinary fuzzy differential inclusion [??] [member of][PHI](t,x), x([t.sub.0])= [x.sub.0] (4) if the control [??]([??]) [member of] LW is fixed and [PHI](t,x)[equivalent to]. F(t, x, [??](t))

The fuzzy differential inclusions (3) has the fuzzy R-solution, if right-hand side of the fuzzy differential inclusion (3) satisfies some conditions [59].

Let X(t) denotes the fuzzy R-solution of the differential inclusion (3), then X(t,w) denotes the fuzzy R-solution of the control differential inclusion (2) for the fixed w([??]) [member of] LW.

Definition 6. The set Y(T)= {X(T,w): w([??]) [member of] LW} be called the attainable set of the fuzzy system (2).

2.2. The some properties of the R-solution

In this section, we consider the some properties of the R-solution of the control fuzzy differential inclusion (1). Let the following condition is true.

Condition A:

A1. A([??]) is measurable on [[t.sub.0],T];

A2. The norm [parallel]A(t)[ parallel] of the matrix A(t)is integrable on [[t.sub.0],T];

A3. The multivalued map W:[[t.sub.0],T] [right arrow] conv ([R.sup.m])is measurable on [[t.sub.0],T];

A4. The fuzzy map G:[R.sub.+] x [R.sup.m] [right arrow] [E.sup.n] satisfies the conditions

1) measurable in t;

2) continuous in w;

A5. There exist v([??]) [member of] [L.sup.2][[t.sub.0],T] and l([??]) [member of] [L.sub.2][[t.sub.0],T] such that |W(t)| [less than or equal to]v(t), |G(t,w)| [less than or equal to]l(t) almost everywhere on [[t.sub.0], T].

A6. The set Q(t)= {G(t,w(t)): w([??]) [member of] LW} is compact and convex for almost every [[t.sub.0],T],i.e. Q(t) [member of] conv([E.sup.n]).

Theorem 3. Let the condition A is true. Then for every w([??]) [member of] LW there exists the fuzzy R-solution X([??],w) such that

1). the fuzzy map X([??],w)has form X(t,w)= [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], where t [member of][[t.sub.0],T]; [PHI](t) is Cauchy matrix of the differential equation [??] = A(t)x;

2). X(t,w) [member of] [E.sup.n] for every t [member of] [[t.sub.0],T];

3). the fuzzy map X([??],w) is the absolutely continuous fuzzy map on ][[t.sub.0],T].

Proof. The proof is easy consequence of the [32,37,40,59] and theorem 1.

Theorem 4. Let the condition A is true. Then the attainable set Y(T) is compact and convex.

Proof. The proof is easy consequence of the [32,37, 40,59] and theorem 1.

We obtained the basic properties of the fuzzy R-solution of systems (1). Now, we consider the some control fuzzy problems.

3. The Optimal Time Problems

Consider the following optimal control problem: it necessary to find the minimal time T and the control w * ([??]) [member of] LW such that the fuzzy R-solution of system (1) satisfies one of the conditions:

X(T,w *)[intersection][S.sub.k] [not equal to] [empty set], (5)

X(T,w *) [subset] [S.sub.k], (6)

X(T,w *) [contains] [S.sub.k], (7)

where [S.sub.k] [member of] [E.sup.n] is the terminal set.

Clearly, these time optimal problems are different from the ordinary time optimal problem by that here control object has the volume.

Definition 6. We shall say that the pair (w *([??]), X([??], w *)) satisfies the maximum principle on [[t.sub.0],T], if there exists the vector-function [psi]([??]), which is the solution of the system

[??] [member of] - [A.sup.T] (t)[psi], [psi](T) [member of] [S.sub.1](0) and the following conditions are true

1) the maximum condition

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

almost everywhere on [[t.sub.0],T];

2) the transversal condition:

a) in the case (5): C([[[X(T, w *)].sup.1], [psi](T))= -C([[S.sub.k].sup.1],-[psi](T));

b) in the case (6): for all [alpha][member of][0,1] C([[X(T, w *)].sup.[alpha], [psi](T))[less than or equal to]C ([[S.sub.k].sup.[alpha]],[psi](T)) and there exists [beta] [member of] [0,1] such that C([[X(T, w *)].sup.[beta]], [psi](T))= C([[S.sub.k].sup.[beta]], [psi](T));

c) in the case (7): for all [beta] [member of] [0,1] C([[X(T, w *)].sup.[alpha]], -[psi](T))[less than or equal] C([[S.sub.k].sup.[alpha]],-[psi](T)) and there exists [beta] [member of][0,1] such that C([[X(T, w *)].sup.[beta]], -[psi](T)) = C([[S.sub.k].sup.[beta]], -[psi](T)).

Clearly, that there cases of the transversal condition of the maximum principle correspond to the three cases of the time optimal problems.

Theorem 5. (necessary optimal condition). Let the condition A are true and the pair (T, w * ([??]))is optimality. Then the pair (w * ([??]),X([??], w *)) satisfies the maximum principle on [[t.sub.0],T].

Proof. The proof is easy consequence of the [32,37, 40].

Example. Consider the following control linear fuzzy differential inclusions

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where x = [([x.sub.1],[x.sub.2]).sup.T] is the state; w=[([w.sub.1],[w.sub.2]).sup.T] [member of] W = [S.sub.1](0)is the control; F [member of] [E.sup.2] is the fuzzy set, where

[upsilon] = [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Consider the following optimal control problem: it is necessary to find the minimal time T and the control w * ([??]) [member of] LW such that the fuzzy R-solution of system satisfies of the conditions:

X(T, w *) [intersection] [S.sub.k] [not equal to] [empty set] where [S.sub.k] [member of] [E.sup.2]is the terminal set such, that

[sigma](x) = [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Q = [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Obviously, the optimal pair T = 2[pi] and w * (t) = (cos(t),-sin(t)) satisfy of the conditions of the theorem 5:

1) (w * (t), [psi](t))= C(W, [psi](t)) for a.e t [member of] [0,2[pi]];

2) C([[X(T, w *)].sup.1], [psi](T)) = -C([[S.sub.k].sup.1], -[psi](T)), where [psi](t) = [(cos(t), -sin(t)).sup.T] for a.e t [member of] [0,2[pi]], [[X (T, w *)].sup.1] = [(T cos(T), -T sin (T)).sup.T] = [2[pi],0).sup.T], [[S.sub.k].sup.1] = {[[x.sub.1], [x.sub.2].sup.T] : [x.sub.1] = 2[pi], -1 [less than or equal to]] [x.sub.2] [less than or equal to] 1}.

doi: 10.4236/iim.2009.13020

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Andrej V. PLOTNIKOV (1), Tatyana A. KOMLEVA (2)

(1) Department of Applied Mathematics, Odessa State Academy of Civil Engineering and Architecture, Odessa, Ukraine

(2) Department of Mathematics, Odessa State Academy of Civil Engineering and Architecture, Odessa, Ukraine Email: {a-plotnikov, t-komleva}@ukr.net

Source Citation

Plotnikov, Andrej V., and Tatyana A. Komleva. "Linear problems of optimal control of fuzzy maps." Intelligent Information Management 1.3 (2009): 139+. Computer Database. Web. 5 Feb. 2010.

Gale Document Number:A216268771

## Friday, February 5, 2010

### Linear problems of optimal control of fuzzy maps.(Report).

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