Dependence on Initial Conditions of Attainable Sets of Control Systems with p-integrable Controls

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1 Nonlinear Analysis: Modelling and Control, 2007, Vol. 12, No. 3, Deendence on Initial Conditions o Attainable Sets o Control Systems with -Integrable Controls E. Akyar Anadolu University, Deartment o Mathematics 26470, Eskisehir, Turkey eakyar@anadolu.edu.tr Received: Revised: Published online: Abstract. Quasi-linear systems governed by -integrable controls, or 1 < < with constraint u ) µ 0 are considered. Deendence on initial conditions o attainable sets are studied. Keywords: control system, integral constraint, attainable set. 1 Introduction In this aer quasi-linear control systems which are nonlinear with resect to hase state vector, linear with resect to control vector and where control inuts are constrained by an integral inequality are studied. It is well known that attainable sets lay an imortant role in control theory. Many roblems o otimization, dynamics, game theory can be stated and solved in terms o attainable sets see 1, 2]). Many roerties o attainable sets or linear and nonlinear systems without integral constraints is well known see 3 5]). On the other hand attainable sets o control systems with -integrable controls are still in interest. General roerties and comutability o attainable sets o latter comletely diers rom ormer see 6 10]). Hence dierent techniques are required. Consider a control system whose behavior is described by a dierential equation ẋt) = t,xt) ) + B t,xt) ) ut), x ) X 0, 1) where x R n is the n-dimensional hase state vector o the system, u R r is the r-dimensional control vector, t,t] < T < ) is the time, t,x) is n-dimensional vector unction, Bt,x) is an n r)-dimensional matrix unction and X 0 R n. The author was suorted by the Anadolu University Research Foundation Project No. AUAF ). 293

2 E. Akyar It is assumed that the realizations ut), t,t], o the control u are restricted by the constraint T ut) dt µ 0, µ 0 > 0, 1 < <, 2) where denotes the Euclidean norm. Inequality 2) describes the constraint on the control ulse. This constraint is used or controls which have limited resources such as uel reserve or jet engines, or caital or economical systems, etc. It is also assumed that the unctions t,x) t,x), t,x) Bt,x) and the set X 0 satisy the ollowing conditions: 1. The set X 0 R n is comact. 2. The unctions t,x) t,x) and t,x) Bt,x) are continuous with resect to t,x) and locally Lischitz with resect to x, that is or any bounded set D,T] R n there exist Lischitz constants L i = L i D) 0, ) i = 1,2) such that t,x ) t,x ) L 1 x x, Bt,x ) Bt,x ) L 2 x x or any t,x ) D, t,x ) D. 3. There exist constants γ i 0, ) i = 1,2) such that t,x) γ x ), Bt,x) γ x ) or every t,x),t] R n. Every unction u ) L,T], R r ), 1 < < ), satisying the inequality 2) is said to be an admissible control, where L,T], R r ) denotes the sace o -ower integrable unctions. By the symbol U we denote the set o all admissible control unctions u ). Let u ) U. The absolutely continuous unction x ):,T] R n which satisies the equation ẋ t) = t,x t)) + Bt,x t))u t) a.e. in,t] and the initial condition x ) = x 0 X 0 is said to be a solution o the system 1) with initial condition x ) = x 0, generated by the admissible control unction u ). By the symbol X,x 0 ) we denote the set o all solutions o the system 1) with initial condition x ) = x 0, generated by all admissible control unctions u ) U and we set X,X 0 ) = { x ) X,x 0 ): x 0 X 0 }, Xt;,X 0 ) = { xt) R n : x ) X,X 0 ) }. The set Xt;,X 0 ) is called the attainable set o the system 1) with constraint 2) at the instant o time t. It is obvious that the set Xt;,X 0 ) consists o all x R n, at which the 294

3 Deendence on Initial Conditions o Attainable Sets o Control Systems with -Integrable Controls solutions o the system 1) which are generated by all ossible controls u ) U arrive at the instant o time t,t]. The calculation o attainable sets can be a tedious task and it is generally treated numerically with the use o a comuter. Thereore it is very imortant to determine how attainable set changes when the initial conditions change. This is studied in Proositions 1 6. The Hausdor distance between the nonemty sets E,F R n is deined as αe,f) = in{r > 0: E F + rb, F E + rb}, 3) where B is unit ball in R n. 2 Preliminaries First, let us give a useul inequality: K1 + K 2 uτ) ) dτ K 1 T ) + K 2 T ) 1 µ0 4) or every u ) U and all t,t], where K 1 and K 2 are ositive constants. Inequality 4) will be used requently in the ollowing sections and it can be easily obtained via Hölder s integral inequality see 11,. 122]). The ollowing roosition states that the grahs o all solutions o the system 1) with constraint 2) is bounded. Proosition 1. The inequality xt) r is ulilled or all x ) X,X 0 ) and t,t], where and q = γ 1 T ) + γ 2 µ 0 T ) 1, d = max { x : x X 0 } r = d + q)exq). 5) Proo. Let x ) X,X 0 ) be any solution o the system 1). Then there exist x 0 X 0 and u ) U such that xt) = x 0 + τ,xτ) ) + B τ,xτ) ) uτ) ] dτ, t t0,t] 295

4 E. Akyar holds. Ater taking the norm o both sides, on using Condition 3 and recalling that d = max{ x : x X 0 }, we obtain, xt) d + γ 1 T ) + γ 1 + γ 2 xτ) uτ) dτ. xτ) dτ + γ 2 In view o Hölder s integral inequality we have uτ) dτ t 1 ) 1 t 1 dτ uτ) dτ By virtue o 6) and since q = γ 1 T ) + γ 2 µ 0 T ) 1 xt) d + q + γ1 + γ 2 uτ) ) xτ) dτ. It ollows rom Gronwall s inequality that t xt) d + q)ex From inequality 4) it ollows that xt) d + q)exq). γ1 + γ 2 uτ) ) ) dτ. uτ) dτ ) 1 µ0 T ) 1. 6) we obtain The right hand side o this last inequality is exactly the number r see 5)). Thus the inequality xt) r holds or all x ) X,X 0 ) and all t,t]. The set Z,X 0 ) = { t,xt) ),T] R n : x ) X,X 0 ) } is called the integral unnel o the system 1) with constraint 2). A corollary o the revious roosition is that the grahs o all solutions o the system 1) is bounded by the cylinder D = { t,x),t] R n : x r }. 7) That is, the inclusion Z,X 0 ) D holds. Here, r > 0 is deined by 5). From now on D will denote the cylinder 7). 296

5 Deendence on Initial Conditions o Attainable Sets o Control Systems with -Integrable Controls 3 Deendence on initial conditions The ollowing roosition determines the deendence o attainable sets on the initial set X 0. Proosition 2. Let X 0 and X 1 be comact subsets o R n. Then the inequality α Xt;,X 0 ),Xt;,X 1 ) ) KαX 0,X 1 ) is valid or all t,t]. Here, K is ositive constant. Proo. Let x 0 ) X,X 0 ) be arbitrary. Then there exist x 0 X 0 and u ) U such that x 0 t) = x 0 + τ,x0 τ) ) + B τ,x 0 τ) ) uτ) ] dτ holds or all t 0,T]. Since X 0 and X 1 are comact subsets o R n, by the deinition o Hausdor distance there exists x 1 X 1 such that x 1 x 0 αx 0,X 1 ) < + holds. Thereore we obtain a new trajectory x 1 ) X,X 1 ) or the system 1) which is generated by the same control u ) U that satisies the initial condition x 1 ) = x 1. Thus we can write x 1 t) = x 1 + τ,x1 τ) ) + B τ,x 1 τ) ) uτ) ] dτ or all t,t]. By Condition 1 we have x 0 t) x 1 t) αx 0,X 1 ) + L1 + L 2 uτ) ) x 0 τ) x 1 τ) dτ or all t,t]. It ollows rom Gronwall s inequality see 11,. 189]) that t x 0 t) x 1 t) αx 0,X 1 )ex L1 + L 2 uτ) ) ) dτ 8) is valid or all t,t]. Taking 4) into account we obtain x 0 t) x 1 t) αx 0,X 1 )ex L 1 T ) + L 2 T ) 1 µ0 ) 297

6 E. Akyar or all t,t]. To shorten notation let us set K = ex L 1 T ) + L 2 T ) 1 µ0 ). 9) Thus we get x 0 t) x 1 t) KαX 0,X 1 ) or all t,t]. The inclusion Xt;,X 0 ) Xt;,X 1 ) + KαX 0,X 1 )B, t,t] 10) is then immediate. Similar arguments yield the inclusion Xt;,X 1 ) Xt;,X 0 ) + KαX 0,X 1 )B 11) or all t,t]. Hence the desired inequality α Xt;,X 0 ),Xt;,X 1 ) ) KαX 0,X 1 ), t,t] is an immediate consequence o 3). Our next result, an easy corollary o the Proosition 2, tells us that the set valued ma X 0 R n Xt;,X 0 ) R n is Lisschitz continuous with Lischitz constant K which is deined by 9). It means that attainable set at any instant o time t continuously deends on the initial set X 0. Proosition 3. Let T > >, X 0,X 1 R n be comact subsets, and r 0 = αx 0,X 1 ) + d 1 ) + d 2 µ 0 ) 1, 12) r = r 0 ex ) L 1 T ) + L 2 µ 0 T ) 1. Then the inequality α Xt;,X 0 ),Xt;,X 1 ) ) r, t,t] holds or the system 1) with constraint 2). Here d 1 and d 2 are ositive constants. Proo. Let t,t] and y 0 Xt;,X 0 ) be arbitrary, then there exist x 0 X 0, x 0 ) X,x 0 ) and u ) U such that y 0 = x 0 t) = x 0 + τ,x0 τ) ) + B τ,x 0 τ) ) uτ) ] dτ 298

7 Deendence on Initial Conditions o Attainable Sets o Control Systems with -Integrable Controls holds. By the deinition o Hausdor distance there exists x 1 X 1 such that x 0 x 1 αx 0,X 1 ). 13) Let x 1 ) X,x 1 ) be solution o the system 1) starting rom the initial oint x 1 X 1 and generated by the same control u ) U as x 0 ), then x 1 t) = x 1 + τ,x1 τ) ) + B τ,x 1 τ) ) uτ) ] dτ is ulilled or all t,t]. Thereore we obtain the inequality, x 0 t) x 1 t) x 0 x 1 + τ,x 0 τ) ) τ,x 1 τ) ) dτ + B τ,x 0 τ) ) B τ,x 1 τ) )] uτ) dτ 14) + 1 ) τ,xτ) + B τ,x0 τ) ) uτ) dτ or all t,t]. From Proosition 1 there exists a cylinder D such that the inclusions Z,X 0 ) D and Z,X 1 ) D holds. Let d 1 = max t,x) D t,x) and d 2 = max t,x) D Bt,x), then it ollows rom 14) and Condition 1 that x 0 t) x 1 t) x 0 x 1 + L1 + L 2 uτ) ) x 0 τ) x 1 τ) ) dτ + 1 d1 + d 2 uτ) ) dτ 15) or all t,t]. In view o 13) and 4) the inequality x 0 t) x 1 t) r 0 + L1 + L 2 uτ) ) x 0 τ) x 1 τ) dτ 299

8 E. Akyar is valid, where r 0 is deined by 12). By virtue o Gronwall s inequality and 4) we ind t x 0 t) x 1 t) r 0 ex L1 + L 2 uτ) ) ) dτ r 0 ex L 1 T ) + L 2 µ 0 T ) 1 ) or all t,t]. Hence the inclusion Xt;,X 0 ) Xt;,X 1 ) + rb holds or all t,t]. Here, r = r 0 exl 1 T ) + L 2 µ 0 T ) 1 ). Similarly choosing an arbitrary element rom Xt;,X 1 ) one can rove that the inclusion Xt;,X 1 ) Xt;,X 0 ) + rb also holds or all t,t]. Thus the desired inequality α Xt;,X 0 ),Xt; ;X 1 ) ) r, t,t] ollows rom 3). An immediate corollary o Proosition 3 is the ollowing. Let X 0 R n and X n R n n = 1,2,...) be comact subsets, αx n,x 0 ) 0 and t n as n. Then the inequality α Xt;t n,x n ),Xt;,X 0 ) ) 0, t,t] holds as n. Let µ 0 and µ 1 be ositive, and U 0 = { u ) L,T], R m ): u ) µ 0 } U 1 = { u ) L,T], R m ): u ) µ 1 }. The set o all solutions and attainable set at instant o time t o the system 1) rom the initial set,x 0 ) which are generated by all controls rom U 0 and U 1 are denoted by X 0,X 0 ), X 0 t;,x 0 ) and X 1,X 0 ), X 1 t;,x 0 ) resectively. The ollowing roosition gives the deendence o attainable sets on the µ

9 Deendence on Initial Conditions o Attainable Sets o Control Systems with -Integrable Controls Proosition 4. Let K > 0 be constant, r 0 = KT ) 1 µ0 µ 1 and r = r 0 1+ ) L 1 T ) + L 2 µ 1 T ) 1 then the inequality ex ) ] L 1 T ) + L 2 µ 1 T ) 1, α X 0 t;,x 0 ),X 1 t;,x 0 ) ) r is ulilled or all t,t]. Proo. Let y 0 X 0 t;,x 0 ) be arbitrary or t,t], then there exist x 0 X 0, x 0 ) X 0,x 0 ) and u 0 ) U 0 such that y 0 = x 0 t) = x 0 + τ,x0 τ) ) + B τ,x 0 τ) ) u 0 τ) ] dτ holds. Let us deine a new control unction u 1 ) via u 0 ) U 0 such that Since u 1 t) = µ 1 µ 0 u 0 t), t,t]. u 1 ) = T u 1 t) dt ) 1 = µ 1 µ 0 T u 0 t) dt ) 1 µ1, we get u 1 ) U 1. We denote the solution o the system 1) starting rom the initial oint,x 0 ) and generated by the control u 1 ) U 1, by x 1 ) X 1,x 0 ) X 1,X 0 ). Setting x 1 t) = y 1, we get y 1 = x 1 t) = x 0 + τ,xτ) ) + B τ,xτ) ) uτ) ] dτ. Hence we obtain the inequality y 0 y 1 + τ,x0 τ) ) τ,x 1 τ) ) dτ B τ,x0 τ) ) u 0 τ) B τ,x 1 τ) ) u 1 τ) dτ. 301

10 E. Akyar It ollows rom Condition 1. y 0 y 1 L1 + L 2 u 0 τ) ) x 0 τ) x 1 τ) dτ + B τ,x1 τ) ) u0 τ) u 1 τ) dτ. Taking K = max t,x) D Bt,x) and using the deinition o the control u 1 ) we clearly have y 0 y 1 L1 + L 2 u 0 τ) ) x 0 τ) x 1 τ) dτ + K 1 µ 1 u 0 τ) dτ, µ 0 where D is deined by 7). From the Hölder s integral inequality it ollows that y 0 y 1 L1 + L 2 u 0 τ) ) x 0 τ) x 1 τ) dτ Let us set + K µ 0 µ 1 T ) 1. r 0 = K µ 0 µ 1 T ) 1. Using Gronwall s inequality and 4) we ind x 0 t) x 1 t) r 0 ex ) L 1 T ) + L 2 µ 0 T ) 1. Deine r = r 0 exl 1 T ) + L 2 µ 0 T ) 1 ), then it ollows that x 0 t) x 1 t) r. Thereore the inclusion X 0 t;,x 0 ) X 1 t;,x 0 ) + rb 16) valid or t,t]. 302

11 Deendence on Initial Conditions o Attainable Sets o Control Systems with -Integrable Controls Similarly, one can obtain the inclusion X 1 t;,x 0 ) X 0 t;,x 0 ) + rb 17) or t,t]. According to inclusions 16) and 17) we obtain the validity o the inequality α X 0 t;,x 0 ),X 1 t;,x 0 ) ) r, t,t] as desired. Let us deine U n = { u ) L,T], R n ): u ) µ n } and denote the set o all solutions and attainable set at instant o time t o the system 1) with initial set,x 0 ) corresonding to control sets U n by X n,x 0 ) and X n t;,x 0 ) resectively. Proosition 4 imlies that or µ n µ 0 as n, the inequality α X n t;,x 0 ),X 0 t;,x 0 ) ) 0 holds as n or all t,t]. By the ollowing roosition it is roved that the set valued ma t Xt;,X 0 ) is Hölder continuous. Proosition 5. For the system 1) with constraint 2) the inequality α X ;,X 0 ),Xt 2 ;,X 0 ) ) M t 2 1 holds or every,t 2,T]. Here, M > 0 is constant. Proo. Without loss o generality we can suose < t 2. Let y 1 X ;,X 0 ) be arbitrary, then there exist x 0 X 0, x ) X,x 0 ) and u ) U such that y 1 = x ) = x τ,x τ) ) + B τ,x τ) ) u τ) ] dτ holds. I we take y 2 = x t 2 ) Xt 2 ;,X 0 ) y 2 = x t 2 ) = x τ,x τ) ) + B τ,x τ) ) u τ) ] dτ. 303

12 E. Akyar is obtained. Thereore we clearly have y 1 y 2 2 τ,x τ) ) t 2 dτ + B τ,x τ) ) u τ) dτ. Let K 1 = max{ t,x) : t,x) D} and K 2 = max{ Bt,x) : t,x) D}, then we ind y 1 y 2 2 K 1 + K 2 ) u τ) dτ Finally, alying Hölder s integral inequality we obtain y 1 y 2 K 1 + K 2 )µ 0 t 2 1. I we set M = K 1 + K 2 )µ 0, then we get y 1 y 2 M t 2 1. Thereore the inclusion X ;,X 0 ) Xt 2 ;,X 0 ) + M t 2 1 B 18) is valid or all,t 2,T]. Similarly, choosing an arbitrary element y 2 rom Xt 2 ;,X 0 ) the inclusion Xt 2 ;,X 0 ) X ;,X 0 ) + M t 2 1 B 19) can be obtained. Combining inclusions 18) and 19) we obtain the desired result. Let E R n. Then diameter o E is denoted by diam E = su x y. x,y E The ollowing roosition gives an uer bound or the diameter o the attainable sets. Proosition 6. Let K = max t,x) D Bt,x) and d = diam X 0, 20) then the inequality diam Xt;,X 0 ) ) d + 2Kµ 0 t ) 1 ex L1 T ) ) holds or all t,t]. 304

13 Deendence on Initial Conditions o Attainable Sets o Control Systems with -Integrable Controls Proo. Let t,t] and y 1,y 2 Xt;,X 0 ) be arbitrary, then there exist x 1 X 0, x 1 ) X,X 0 ), u 1 ) U such that y 1 = x 1 t) = x 1 + τ,x1 τ) ) + B τ,x 1 τ) ) u 1 τ) ] dτ holds and there exist x 2 X 0, x 2 ) X,X 0 ), u 2 ) U such that y 2 = x 2 t) = x 2 + τ,x2 τ) ) + B τ,x 2 τ) ) u 2 τ) ] dτ is valid. It ollows rom Condition 1 and 20) that y 1 y 2 x 1 x 2 + L 1 + d + L 1 x 1 τ) x 2 τ) dτ B τ,x1 τ) ) u1 τ) dτ + x 1 τ) x 2 τ) dτ + K B τ,x2 τ) ) u2 τ) dτ t In accordance with Hölder s integral inequality we obtain y 1 y 2 d + L 1 u 1 τ) dτ + x 1 τ) x 2 τ) dτ + 2Kµ 0 t ) Thereore utilizing the Gronwall s inequality see 11,. 189]) we ind y 1 y 2 ) d + 2Kµ 0 t ) 1 ex L1 T ) ). Since t,t] and y 1,y 2 Xt;,X 0 ) arbitrary, we ind diam Xt;,X 0 ) ) d + 2Kµ 0 t ) 1 ex L1 T ) ) or all t,t]. 1 u 2 τ) dτ It is clear rom Proosition 6 that diam Xt;,X 0 ) diam X 0 as t. We conclude rom Proositions 1 6 that attainable set o the system 1) with constraint 2) at the instant o time t,t] continuously deends on initial set X 0 and µ 0. Besides, the set valued mas X 0 Xt;,X 0 ) and t Xt;,X 0 ) are Lischitz and Hölder continuous resectively. ]. 305

14 E. Akyar Reerences 1. F. L. Chernousko, State Estimation or Dynamic Systems, CRC Press, Boca Raton, Florida, E. V. Grigorieva, E. N. Khailov, Attainable Set o a Nonlinear Controlled Microeconomic Model, J. Dyn. Control Syst., 112), , A. M. Formalsky, A. N. Sirotin, On the Geometric Proerties o Reachable and Controllable Sets or Linear Discrete Systems, J. Otim. Theory Al., 1222), , H. Hermes, J. P. Lasalle, Functional Analysis and Time Otimal Control, Academic Press, New York, E. B. Lee, L. Markus, Foundations o Otimal Control Theory, Wiley, New York, Kh. G. Guseinov, A. A. Neznakhin, V. N. Ushakov, Aroximate construction o reachable sets o control systems with integral constraints on the controls, Nonlinear Anal., 634), , Kh. G. Guseinov, O. Ozer, E. Akyar, On the continuity roerties o the attainable sets o control systems with integral constraints on control, Nonlinear Anal., 563), , H. W. Lou, On the Attainable Sets o Control Systems with -Integrable Controls, J. Otim. Theory Al., 1231), , M. Motta, C. Sartori, Minimum time with bounded energy, minimum energy with bounded time, SIAM J. Control Otim., 423), , B. T. Polyak, Convexity o the reachable set o nonlinear systems under L 2 bounded controls, Dyn. Contin. Discrete Imuls. Syst., Ser. A, Math. Anal., 112 3), , J. Warga Otimal Control o Dierential and Functional Equations, Academic Press, New York, P. Collins, Continuity and Comutability o Reachable Sets, Theoret. Comut. Sci., ), ,