Extension of a Problem of the Game Control

Extension of a Problem of the Game Control Chentsov A.G. ∗ Shapar Ju.V. ∗∗ ∗ Institute of Mathematics and Mechanics, Ural Branch of Russian Academy of Sci., Ekaterinburg, Russia (Tel: (343)374-25-81; e-mail: chentsov@ imm.uran.ru ) ∗∗ Institute of Mathematics and Mechanics, Ural Branch of Russian Academy of Sci., Ekaterinburg, Russia (Tel: (343)374-25-81; e-mail: [email protected])

Abstract: The terminal game problem of control of the linear system with a discontinuity in coefficients under controlling influences and impulse constraints is considered. In the class of finitely additive measures with the property of the weak absolute continuity with respect to the restriction of the Lebesgue measure to some ”sufficient” measurable structure, the generalized game problem of control is constructed. Keywords: maximin, finitely additive measure, net, generalized control 1. INTRODUCTION Constructions of extensions are used in theory of control very widely. We note the investigations of J.Warga, R.V.Gamkrelidze, A.D.Ioffe and V.M.Tikhomirov connected with the classical theory of control. For game problems of control, different generalized problems are used by N.N.Krasovskii and A.I.Subbotin. Under the proof of fundamental theorem about the alternative of N.N.Krasovskii and A.I.Subbotin, the generalized elements were used in the important definition of the bridges stability of N.N.Krasovskii. In the following, the approach connected with the employment of real-valued finitely-additive measures (FAM) [Danford (1962)] for extensions of the control problems is used. The basic reason such employment of FAM is construction of extension is connected with the known Alaoglu theorem about conditions of a compactness in the sense of the ∗-weak topology. Of course, we take in account the possibility of the identifying of FAM with the bounded variation and linear continuous functionals on discontinuous functions. As a resalt, we obtain the natural duality of Banach spaces. We use FAM as generalized controls. In fact, this means that linear continuous functionals are used as generalized controls. So, the distinctive combination of functional analysis and control theory is used. In addition, the employment of FAM is very essential; countably additive measures are insufficient. Of course, we obtain nonmetrizable topological spaces of generalized controls formulizable as FAM. But this spaces are compactums (this property is realized by Alaoglu theorem). Of course, we keep in mind the corresponding subspace of the space of FAM with the bounded variation equipping with the ∗-weak topology. In our setting, the very general variants of the sets of usual controls are assumed. The corresponding variants of the sets of generalized controls in the space of FAM are realized by the ∗-weak closure. In addition, it is im-

portant that this operation of the closure can be define constructively. Therefore such cases are indicated. These cases are established in the following investigations: [Chentsov (1996,1)], [Chentsov and Morina (2002)], [Chentsov (1999)], [Chentsov (1998)]. In the following we use only real-valued FAM. In the problems of control are used very often. We note some investigations in which (for extension constructions) vector FAM are used; see [Chentsov (1997)], [Chentsov (2001)], [Chentsov (1996,2)]. In connection with general questions of the finitely additive measures theory, we note the know monograph [Rao (1983)]; moreover, see [Danford (1962), ch.III,IV]. In connection with questions of extensions in control problems, we note monographs [Warga (1972)] and [Gamkrelidze (1977)]. We consider the following linear system: x(t) ˙ = u(t)α(t) + v(t)β(t),

(1)

where t ∈ I0 , [t0 , θ0 ] (t0 < θ0 ) and x(t) ∈ Rn . Here n ∈ N , where N , {1; 2; 3; ...}; suppose that u = u(·), v = v(·), where u(·) = (u(t) ∈ R, t0 ≤ t < θ0 ), v(·) = (v(t) ∈ R, t0 ≤ t < θ0 ) are controls; the functions α = α(·) and β = β(·) are defined on I , [t0 ; θ0 [, α : I → Rn , β : I → Rn ; these functions α and β are fixed and may be discontinuous. For the simplicity, now we suppose that u and v are piece wise constant and continuous from the right real-valued function on I. Now, we use an informative variant of the account. We fix x0 ∈ Rn and suppose (in the following) that x(t0 ) = x0 ; see (1). Finally, we fix a continuous function f0 from Rn into R.

We note that the system (1) can be obtained by the known (see [Krasovskii (1974)]) nonsingular linear transformation of a linear system of the more general type. We fix a continuous function f0 from Rn into R as the goal function for a terminal criterion. If u and v are controls in (1) and xu,v = xu,v (·) is the trajectory (of our system) generated by u and v, then the value f0 (xu,v (θ0 )) ∈ R is defined. We fix two nonempty sets U and V of controls u = u(·) and v = v(·) respectively; suppose that U and V satisfy a natural resource constraints. Our criterion is defined by the rule (u, v) 7→ f0 (xu,v (θ0 )) : U × V → R; we consider the natural value of the game problem: sup inf f0 (xu,v (θ0 )) ∈ R. (2)

respect to a finitely additive measure µ ∈ A(L). We fix two nonempty sets U, U ⊂ B0 (I, L), and V, V ⊂ B0 (I, L). Suppose that     Z Z  |u|dλ ≤ cU ∀u ∈ U  &  |v|dλ ≤ cV ∀v ∈ V  ,(4) I

I

where cU ≥ 0, cV ≥ 0. In Section 1, the particular case was considered. If f ∈ B0 (I, λ), then f ∗ λ is the indefinite λ−integral of f ; so f ∗ λ is defined by traditional rule [Chentsov (1996,1), p.69] Z L 7→

v∈V u∈U

In the following, the extension of this game problem is considered. In addition, for (2), the generalized representation in the form of the standard maximin of a continuous functional on the product of two compactums is constructed. We construct these compactums in the space of finitely additive measures on I with the form of a semi-algebra of sets. 2. THE EXTENSION IN THE CLASS OF FINITELY ADDITIVE MEASURES In the following, we consider a more general case of the informative problem of Section 1. For this, we fix an arbitrary semi-algebra L of subsets of I. Suppose that the following conditious are satisfied: 1)[a, b[∈ L for any a ∈ I0 and b ∈ I0 ; 2) all sets of L are measurable by Borel. We denote by λ the ”usual” contraction of the Lebesgue measure on L. The set of all real-valued FAM on L with the bounded variation is denoted by A(L). The linear space A(L) is generated by the nonnegative cone (add)+ [L] of all real-valued nonnegative FAM on L. Let B0 (I, L) be the linear spane of all indicators (see [Neveu (1969)]) of sets of L. So, B0 (I, L) is the space of L−step-function. Let B(I, L) is the closure of B0 (I, L) in the space B(I) of all bounded real-valued function on I with the traditional sup-norm k·k. Of course, B(I, L) with the norm induced from B(I, k · k) is the Banach space too. The spaces B ∗ (I, L) and A(L) are identified (see [Danford (1962), ch.IV] and [Chentsov (1996,1), §3.4]). We equip A(L) with the standard ∗−weak topology τ∗ (L) to the duality (B(I, L), A(L)). Then (A(L), τ∗ (L)) (3) is a locally convex σ−compactum. We use the operation of the ∗−weak closure of the images of subsets of B0 (I, L) under the transformation on the base of the indefinite λ−integral; see [Chentsov (1996,1), §3.4].

f dλ : L → R. L

We note the important property: the space B(I, L) is a Banach algerba. Therefore, f g ∈ B(I, L) ∀f ∈ B(I, L) ∀g ∈ B(I, L). Of course, we can consider the following integrals: Z f gdλ ∈ R ∀f ∈ B(I, L) ∀g ∈ B(I, L). I

In addition see [Chentsov (1996,1), p.69] for g ∈ B(I, L) and h ∈ B0 (I, L), Z Z ghdλ = gd(h ∗ λ). I

I

On this basis, we define the natural immersion of the set U in the TS (3) by the following rule: u 7→ u ∗ λ : U → A(L). Analogously we use the immersion v 7→ v ∗ λ : V → A(L) of the set V in the TS (3). So, we consider L-step-functions as FAM. But, in the capacity of generalized controls we use not only such FAM. Namely, we use the corresponding limits of the above-mentioned simplest general controls (generated by L-step-functions). In addition, we use the limit passage in the TS (3). For this limit realization of FAM with the bounded variations, the employment only sequences is insufficient. For the construction of the corresponding sets in the TS (3), the employment of nets is essentially. Let e , cl({u ∗ λ : u ∈ U }, τ∗ (L)), U Ve , cl({v ∗ λ : v ∈ V }, τ∗ (L)), where cl(·, τ∗ (L)) is the closure operator in the space (3). e and Ve are the nonempty compuctums in the sense Then, U e ⊂ Aλ [L] and Ve ⊂ Aλ [L], where of (3); in addition, U

Of course, we understand the ∗−weak closure as the closure in the topological space (3). In the following we use the designation cl(·, τ∗ (L)) for the closure operation in (3). We recall that the compactness conditions in (3) is defined by the known Alaoglu theorem.

Aλ [L] , {µ ∈ A(L)|∀L ∈ L (λ(L) = 0) ⇒ (µ(L) = 0)}. We note that, for many practically interesting variants of e and Ve can be constructed. In this U and V, the sets U connection, we recall the known statement (see [Chentsov (1996,1), p.86] and [Chentsov (1997), p.53]): Aλ [L] = cl({f ∗ λ : f ∈ B0 (I, L)}, τ∗ (L)). It is obvious that the following inclusion take place: e ⊂ Aλ [L], Ve ⊂ Aλ [L]. U

In the following we use simplest definition (see [Chentsov (1996,1), ch.3]) for integral of a function f ∈ B(E, L) with

So, usual controls are immersed in the space Aλ [L]. e and ν ∈ Ve are used as Finitely additive measures µ ∈ U

generalized controls. Suppose that all components of the vector-function α and β are contained in B(I, L). Then, for u ∈ U and v ∈ V, the function ϕu,v from I0 into Rn is defined by the rule Z Z v(τ )β(τ )λ(dτ ). u(τ )α(τ )λ(dτ ) + ϕu,v (t) , x0 +

and the value (the maximin) e V , max min Φ(µ, ν) = max Ψ(ν) e µ∈Ue e ν∈V ν∈V are defined. The following property takes place. Theorem 1. The equality V and γ is valid:

[t0 ,t[

[t0 ,t[

So, usual trajectories are introduced; suppose that Φ is the functional on U × V for which Φ(u, v) , f0 (ϕu,v (θ0 )). By [Chentsov (1996,1)] the boudedness property of Φ follows. We investigate the value γ , sup inf Φ(u, v).

V = γ. Now, we consider the brief scheme of the proof. First, we note that the set {u ∗ λ : u ∈ U } is every where dense e , τe∗ (L)). Analogously, {v ∗ λ : v ∈ V } is everywhere in (U U e is the continuous functional, dense in (Ve , τeV∗ (L)). Since Φ the equalities

v∈V u∈U

e and ν ∈ Ve , we introduce the generalized Under µ ∈ U trajectory ϕ eµ,ν on I0 by the rule Z Z ϕ eµ,ν (t) , x0 + α(τ )µ(dτ ) + β(τ )µ(dτ ); [t0 ,t[

e e ∗ λ, ν). min Φ(µ, ν) = inf Φ(u u∈U e µ∈U

(6)

On the other hand, for u ∈ U and v ∈ V the equality ϕu,v = ϕ eµ,ν

(7)

[t0 ,t[

of course, ϕ eµ,ν is the function from I0 into Rn . In terms of this functions, the criterion of the generalized problem is constructed: the functional e :U e × Ve → R Φ (5)

holds, where µ = u ∗ λ and ν = v ∗ λ. So, by (7) usual trajectories are imbedded in the space of generalized e trajectories. From (7), the equality Φ(u, v) = Φ(µ, ν), where µ and ν are generated by u and v respectively. Therefore, for v ∈ V

is defined by the following rule: e Φ(µ, ν) , f0 (ϕ eµ,ν (θ0 )). By the above-mentioned properties of the indefinite integral the following useful property is realized. Namely, if u ∈ U and v ∈ V ; then the FAM (µ = u ∗ λ) & (ν = v ∗ λ) realize the two equalities ϕu,v = ϕ eµ,ν . e and Ve with the topologies We equip the nonempty sets U of the corresponding subspaces of the space (3). As a result, we obtain the two compactums e , τeU∗ (L)) and (Ve , τeV∗ (L)), (U

where τeU∗ (L) and τeU∗ (L) are the topologies induced of (3). Therefore, the natural topology τeU (L)) ⊗ τeV (L) e × Ve realizes the compactum of the set U

e × Ve , τe∗ (L) ⊗ τe∗ (L)) (U U V corresponding to the topological product of the abovementioned compactums. e (5) is a continuous functional on this In addition, Φ e × Ve , τe∗ (L) ⊗ τe∗ (L)). As a corollary, Φ(·, e ν) compactum (U U V ∗ e is a continuous functional on (U , τeU (L)) for any ν ∈ Ve . Moreover, the functional Ψ of the form e ν 7→ min Φ(µ, ν) : Ve → R e µ∈U is continuous on (Ve , τeV∗ (L)). Now, we consider the genere For alized problem of game control with the criterion Φ. this problem, the nonempty extremal set e e Veopt , {ν0 ∈ Ve | min Φ(µ, ν) ≤ min Φ(µ, ν0 ) ∀ν ∈ Ve } e e µ∈U µ∈U

e ∗ λ, ν) = inf Φ(u, v), inf Φ(u

u∈U

u∈U

(8)

where ν = v ∗ λ. Of course, we can use (6) in the case ν = v ∗ λ. Then, by (6) and (8) e v ∗ λ) = Ψ(v ∗ λ), inf Φ(u, v) = min Φ(µ, e µ∈U

u∈U

(9)

where v ∈ V. As a corollary, for the ”usual value” γ, we obtain the equality e v ∗ λ). γ = sup min Φ(µ, v∈V µ∈U e

(10)

But, v ∗λ ∈ Ve by the definition of Ve . Therefore, from (10), γ ≤ V.

(11)

Recall that Veopt 6= ∅. We choose arbitrarily ν0 ∈ Ve . Then, Ψ is a continuous functional in the point ν0 . Moreover, any neighborhood of ν0 in τeV∗ (L) contains some point v ∗ λ, v ∈ V. As a corollary, for any ε ∈]0, ∞[, we can indicate a neighborhood N of ν0 in the above-mentioned sense for which |Ψ(ν) − Ψ(ν0 )| = |Ψ(ν) − V| < ε ∀ν ∈ N. Later, we choose vN ∈ V such that vN ∗ λ ∈ N ; therefore, |Ψ(vN ∗ λ) − V| < ε. With the employment of (9), we obtain (in particular) the inequality V − ε < inf Φ(u, v). u∈U

By the definition of γ we obtain the inequality V − ε < γ. But, the choice of ε was arbitrary. Therefore, V ≤ γ. Using (11), we obtain the statement of our Theorem. We note that Veopt is a closed set is (Ve , τeV∗ (L)).

3. THE QUESTION ABOUT A STABILITY

100 ) V = {v ∈ B0 (I, L) |

R

|v|dλ ≤ b} and Ve = {ν ∈

I

In the previous Section, we consider the game problem with the fixed goal function f0 . Therefore, e = Φ[f e 0 ], Ψ = Ψ[f0 ], V = V[f0 ], Veopt [f0 ]. Φ Now, we investigate the question of the stability under a variation of f0 . We introduce the usual attainability domain G , {ϕu,v (θ0 ) : u ∈ U, v ∈ V } and the closure G of this domain. Proposition 1. The following equality is valid: e , ν ∈ Ve }. G = {ϕ eµ,ν (θ0 ) : µ ∈ U For any continuous function f on Rn , we introduce ef , Ψ e f , Veopt,f , and V with the replacement f0 → f. Φf , Φ The proximity of f and f0 is defined by the sup-norm of the restrictions (f | G) and (f0 | G) (of the functions f and f0 respectively) to the set G. Then, the dependence f 7→ Vf : C(Rn ) → R is continuous in the point f0 . Moreover, the set-valued dependence f 7→ Veopt,f is upper semi-continuous in the point f0 . This property has the sense of a stability. Now, we refine the corresponding statement. Consider the space B(G) of all real-valued bounded functions on G. We equip B(G) with the traditional sup-norm · . Then, for any f ∈ C(Rn ) t

t

(f | G) = (f (x))x∈G ∈ B(G). In addition, |Ψf1 (ν) − Ψf2 (ν)| ≤ (f1 | G) − (f2 | G) ∀f1 ∈ C(Rn ), ∀f2 ∈ C(Rn ), ∀ν ∈ Ve . As a corollary, t

t

|Vf1 −Vf2 | ≤ (f1 | G−(f2 | G) ∀f1 ∈ C(Rn ) ∀f2 ∈ C(Rn ). Theorem 2. If f0 ∈ C(Rn ), G0 ∈ τeV∗ (L), and Veopt,f0 ⊂ G0 , then there exist δ0 ∈]0, ∞[ such that ∀f ∈ C(Rn ) ( (f | G) − (f0 | G) < δ0 ) ⇒ (Veopt,f ⊂ G0 ). t

t

t

t

Of course, in Theorem 2, we obtain the property of the stability of generalized problem. 4. SOME EXAMPLES e ) and We note several concrete variants of pairs (U, U e (V, V ). For this, we introduce some new designations. If µ ∈ A(L), then vµ (I) is the total variation of µ on the set I. By B + (I, L) we denote the set of all nonnegative functions of B0 (I, L). Fix a ∈ [0, ∞[ and b ∈ [0, ∞[. In this terms, we can realize the following concrete variants e ) and (V, Ve ) : of (U, U R 0 e = {µ ∈ 1 ) U = {u ∈ B0 (I, L) | |u|dλ ≤ a} and U I

Aλ [L] | vµ (I) ≤ a}; R e = {µ ∈ 20 ) U = {u ∈ B0+ (I, L) | udλ ≤ a} and U I

(add)+ [L; λ] | µ(I) ≤ a}; R e = {µ ∈ 30 ) U = {u ∈ B0+ (I, L) | udλ = a} and U I

(add)+ [L; λ] | µ(I) = a};

Aλ [L] | vν (I) ≤ b}; R 200 ) V = {v ∈ B0+ (I, L) | vdλ ≤ b} and Ve = {ν ∈ I

(add)+ [L; λ] | ν(I) ≤ b}; R 300 ) V = {v ∈ B0+ (I, L) | vdλ = b} and Ve = {ν ∈ I

(add)+ [L; λ] | ν(I) = b}. e ) and (V, Ve ) 100 ) − 300 ) We can combine the pair (U, U arbitrarily. For example, we can use the combination 10 ) and 300 ). Of course, we can indicate other variants; see for example, constructions in [Chentsov (1997)]. 5. CONCLUSION For the linear system of control, extension in the class of FAM is constructed. In the corresponding generalized problem the values of maximum and minimum operations are reached (in the class of FAM with the property of weakly utterly continue with the respect of the restriction of the Lebegues measure). The generalized problem is stable by result under variations of the goal function in the initial problem. REFERENCES N.N.Krasovskii, A.I.Subbotin Positional difference games. Nauka, 456: page 161, Moskow, 1974 J.Neveu Mahtematical bases of probability theory. Mir, 309, Moskow, 1969. N.Danford, J.T.Shvarts Linear operators. Theory. Nauka, 895, Moskow, 1962. A.G.Chentsov Finitely additive measures and relaxations of extremal problems. Plenum Publishing Corporation, 244, New York, London and Moscow, 1996,1 A.G.Chentsov Asymptotic attainability. Kluwer Publishers, 322, Dordrecht-Boston-London, 1997 A.G.Chentsov and S.I.Morina Extension and relaxations. Kluwer Academic Publishers, 408: Dordrecht-BostonLondon, 2002 A.G.Chentsov On the question of correct extension of a problem on the choice of the probability density under restrictions on a system of mathematical expectation. Usp. mat. nauk,V.50, N5(305): P.223-242, 1999 (Russian) A.G.Chentsov Universal properties of generalized integral constraints in the class of finitely additive measures. Functional Differential Equations, N1-2, P.69-105, 1998 A.G.Chentsov Finitely Additive Measures and Extension Constructions. Atti Semin. Mat. Fis., V.49., P.531-545, Univ.Modena, 2001 A.G.Chentsov Vector finite-additive measures and the regularization problem of the construction of the sets of asymptotic attainability. Trudy IMM UrO RAN, V.4, P.266-295. Ekaterinburg, 1996,2 Bhaskara Rao K.P.S.,Bhaskara Rao M. Theory of charges. A study of finitely additive measures. Acad. Press, 253. New York, 1983 J.Warga Optimal control of differential and functional equations. Acad. Press, New York, 1972 R.V.Gamkrelidze Foundations of optimal control theory. Izdat.Tbil.Univ., 254. Tbilissi, 1977 (Russian)