On the continuity property of L p balls and an application

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1 J. Math. Anal. Al ) On the continuity roerty of L balls and an alication Kh.G. Guseinov, A.S. Nazliinar Anadolu University, Science Faculty, Deartment of Mathematics, 6470 Eskisehir, Turkey Received 8 November 006 Available online 0 February 007 Submitted by M.A. Noor Abstract In this aer continuity roerties of the set-valued ma B ),, + ), are considered where B ) is the closed ball of the sace L [t 0,θ]; R m ) centered at the origin with radius. It is roved that the set-valued ma B ),, + ), is continuous. Alying obtained results, the attainable set of the nonlinear control system with integral constraint on the control is studied. The admissible control functions are chosen from B ). It is shown that the attainable set of the system is continuous with resect to. 007 Elsevier Inc. All rights reserved. Keywords: L sace; Set-valued ma; Control system; Attainable set. Introduction Let be Euclidean norm in R m, u ) <+ ) be a norm in L [t 0,θ],R m ), where L [t 0,θ],R m ) denotes the sace of measurable functions u ) : [t 0,θ] R m with bounded u ) norm and u ) = θ t 0 ut) dt). Suorted by the Scientific and Technological Research Council of Turkey TUBITAK) roject No. 06T0. * Corresonding author. Fax: addresses: kguseynov@anadolu.edu.tr Kh.G. Guseinov), asnazliinar@anadolu.edu.tr A.S. Nazliinar) X/$ see front matter 007 Elsevier Inc. All rights reserved. doi:0.06/j.jmaa

2 348 Kh.G. Guseinov, A.S. Nazliinar / J. Math. Anal. Al ) For and > 0weset B ) = u ) L [t0,θ],r m) : u )..) It is obvious that B ) is the closed ball centered at the origin with radius in L [t 0,θ],R m ). The ausdorff distance between the sets A R m and E R m is denoted by ha, E) and is defined as ha, E) = max su dx,e),su dy,a) x A y E where dx,e) = inf x y : y E. The ausdorff distance between the sets U L [t 0,θ],R m ) and V L [t 0,θ],R m ) is denoted by h U, V ) and is defined as h U, V ) = max su x ) V ) ) d x ), U, su d y ), V y ) U where d x ), U) = inf x ) y ) : y ) U, [, ), [, ). For Ω R n we denote by μω) the Lebesgue measure of the set Ω. The need to evaluate the distance between the sets arises in various roblems of theory and alications see, e.g., [,,4 6,8 0,3] and references therein). In this aer, the ausdorff distance between the sets B ) and B ) is studied where > and >. In Section we rove that h B ), B )) 0as 0 Proosition.5). In Section 3 it is shown that h B ), B )) 0as + 0 Proosition 3.5). As a corollary of Proositions.5 and 3.5, Theorem 3.6 concludes that h B ), B )) 0as. In Section 4 we consider attainable sets of the nonlinear control system with integral constraints on control. B ) is chosen as the set of admissible control functions. As an alication of Theorem 3.6, it is roved that the attainable set of the control system is continuous with resect to Theorem 4.). Let 0, ). Weset B ) = u ) B ): ut) for every t [t0,θ]. The following roosition characterizes the ausdorff distance between the sets B ) and B ). Proosition.. Let >, >0. Then the inequality h B ), B ) ) holds. Proof. Let us choose an arbitrary u ) B ) and define a function u ) : [t 0,θ] R m,setting for t [t 0,θ] ut), ut), u t) = ut) ut), ut) >..)

3 Kh.G. Guseinov, A.S. Nazliinar / J. Math. Anal. Al ) It is not difficult to verify that u ) B ).Let Ω = τ [t 0,θ]: uτ) >. Then using ölder and Minkowski inequalities, we have from.) that u ) u ) = ut) u t) dt μω)..3) Ω Since u ) B ) and uτ) > for every τ Ω, we obtain μω) uτ) dτ θ uτ) dτ μ 0 and consequently Ω t 0 μω) μ 0. Then it follows from.3) and.4).4) u ) u ) ) μ 0 = μ 0. Since u ) B ) is arbitrarily chosen, we get the inequality su d u ), B ) ) μ 0..5) u ) B ) Since B ) B ) then.5) comletes the roof of the roosition. We obtain the following corollary from Proosition.. Corollary.. Let > and ε>0. Then there exists ε) > such that for all > ε) the inequality h B ), B ) ) ε holds for any [ +, ].. Left evaluation of B ) In this section, we will evaluate the ausdorff distance between the sets B ) and B ) as 0. Let μ0 α 0 = min,..)

4 350 Kh.G. Guseinov, A.S. Nazliinar / J. Math. Anal. Al ) Proosition.. Let >, ε 0,α 0 ) and > >. Then there exists δ = δ ε,, ) 0, ] such that the inclusion B ) B ) + θ t 0 )εb ) holds for all δ, ) where α 0 > 0 is defined by.) and B ) is defined by.). Proof. Let δ ε,, ) = min β ε, ), β ε, ),, ) where +, ) = max,, + log μ 0 ) β ε, ) =, + log +ε μ 0 ) β ε, ) =. + log ε μ0 ε It is not difficult to verify that δ ε,, ) 0, ]. Let u ) B ) be arbitrarily chosen and δ ε,, ), ).Weset u t) = u t) u t) μ 0, t [t 0,θ]..) Since u ) B ) and δ ε,, ), ), it is ossible to rove that u ) B ). Denote Aε) = t [t 0,θ]: 0 u t) ε, Bε)= t [t 0,θ]: ε< u t). Let t Aε). Then 0 u t) ε. Since ε< and < then we obtain u t) ) u t) ε.3) for every t Aε). Let t Bε). Then ε< u t) and this gives ) u t) ) ) ε..4) Since δ ε,, ), ), then we get from.4) that the inequality ) u t) μ ε 0 holds for every t Bε) and consequently

5 Kh.G. Guseinov, A.S. Nazliinar / J. Math. Anal. Al ) u t) u t) ) ε..5) Finally, it follows from.3) and.5) that u ) u ) [ ε μ Aε) ) + μ Bε) )] εθ t 0 )..6) Since δ ε,, ), ) and u ) B ) are arbitrarily chosen,.6) imlies the validity of the roosition. From Corollary. and Proosition. the validity of the following roosition follows. Proosition.. Let >, ε 0,α 0 ) where α 0 > 0 is defined by.). Then there exists γ = γ ε) 0, ] such that the inclusion B ) B ) + εb ) holds for any γ, ). Proof. By Corollary. there exists ε) > such that for every > ε) the inclusions B ) B ) + ε 3 B ), B ) B ) + ε 3 B ).7) hold for any [ +, ]. Let ε) = ε), ε) = 3 ε). Then by virtue of Proosition. there exists γ ε) = δ ε, ε), ε)) 0, ] such that the inclusion B ε) ) B ε) ) + ε 3 B ).8) holds for all γ ε), ). The roof of the roosition follows from.7) and.8). Now, let us give an uer estimation of the set B ) as 0. Let [ + μ = max μ0 :, ],.9) L = + μ )θ t 0 )..0) Proosition.3. Let >, ε 0,α 0 ), > where α 0 > 0 is defined by.). Then there exists δ = δ ε, ) 0, ] such that the inclusion B ) B ) + L ε B ) holds for any δ, ). Proof. Let δ ε, ) = min β 3 ε, ), β 4 ε, ) where

6 35 Kh.G. Guseinov, A.S. Nazliinar / J. Math. Anal. Al ) β 3 ε, ) = min log ε β 4 ε, ) = min log + ε, ε, It is obvious that δ ε, ) 0, ]. Let δ ε, ), ). Now let us choose an arbitrary u ) B ) and define a function u ) : [t 0,θ] R m by setting u t) = ut) ut), t [t 0,θ]..), Since u ) B ) one can show that u ) B ). Now let us set Aε) = t [t 0,θ]: 0 ut) ε, Bε)= t [t0,θ]: ε< ut). Let t Aε). Then 0 ut) ε. Since ε<α 0 and δ ε, ), ) [ +, ) we get from.) ut) u t) dt εθ t 0 ) + μ0 ε θ t0 ) Aε) εθ t 0 ) + ε μ θ t 0 ) ε + μ )θ t 0 ).) where μ is defined by.9). Let t Bε). Then ε< ut) and. ) ) μ0 μ0 ε ut) Since δ ε, ), ) then.3) imlies ) μ0 ut) ε and consequently μ0 )..3) ut) ) μ0 ut) ε.4) for every t Bε). Thus, it follows from.) and.4) u ) u ) ε θ t0 ) [ + μ + ε ] ε θ t0 )[ + μ ]=ε L where L is defined by.0). Since δ ε, ), ) and u ) B ) are arbitrarily chosen, we obtain the validity of the roosition. The following roosition gives an uer estimation of the set B ) as 0.

7 Kh.G. Guseinov, A.S. Nazliinar / J. Math. Anal. Al ) Proosition.4. Let >, ε 0,α 0 ) where α 0 > 0 is defined by.). Then there exists γ = γ ε) 0, ] such that the inclusion B ) B ) + εb ) holds for any γ, ). Proof. By Corollary. there exists ε) > such that for all > ε) the inclusions B ) B ) + ε 3 B ), B ) B ) + ε 3 B ).5) hold for any [ +, ]. Let ε)= ε). Then due to Proosition.3 there exists δ ε) = δ ε, ε)) 0, ] such that the inclusion B ε) ) B ε) ) + ε 3 B ).6) holds for any δ ε), ). Let γ = γ ε) = δ ε). Then.5) and.6) comlete the roof. From Proositions. and.4 we get the following roosition. Proosition.5. Let >, ε 0,α 0 ) where α 0 > 0 is defined by.). Then there exists δ = δ ε) 0, ] such that the inequality h B ), B ) ) ε holds for all δ, ). 3. Right evaluation of B ) In this section we will study right continuity of the set-valued ma B ),, + ). The following roosition gives an uer estimation of the set B ) as + 0. Proosition 3.. Let >, ε 0,α 0 ), > > where α 0 > 0 is defined by.). Then, there exists ν = ν ε,, ) 0, ] such that the inclusion B ) B ) + θ t 0 )εb ) holds for any, + ν ). Proof. Let ν ε,, ) = min β ε,, ), β ε,, ),, ) where ), ) = min, + log, β ε,, ) = min log + ε,,

8 354 Kh.G. Guseinov, A.S. Nazliinar / J. Math. Anal. Al ) β ε,, ) = min log ε μ0 ε,. It is obvious that ν ε,, ) 0, ]. Let, + ν ε,, )) and choose an arbitrary u ) B ). Define a function u ) : [t 0,θ] R m by setting u t) = ut) ut), t [t 0,θ]. 3.) It is not difficult to show that u ) B ). Let us denote Aε) = t [t 0,θ]: 0 ut) ε, Bε)= t [t 0,θ]: ε< ut). Let t Aε). Since ε< and > then we obtain ut) ) ut) μ ε. 3.) 0 Let t Bε). Then ε< ut) and consequently ) ) ) ut) ε. 3.3) Since, + ν ε,, )), from 3.3) we see that the inequality ) ut) μ ε 0 is satisfied and hence ut) ut) ) Thus, from 3.) and 3.4) we obtain the inequality u ) u ) εμ Aε) ) + εμ Bε) ) εθ t 0 ). ε. 3.4) Since, + ν ε,, )) and u ) B ) are arbitrarily chosen, the roof is comleted. Proosition 3.. Let >, ε 0,α 0 ) where α 0 > 0 is defined by.). Then there exists γ = γ ε) 0, ] such that the inclusion B ) B ) + εb ) holds for any, + γ ). The roof of Proosition 3. follows from Corollary. and Proosition 3.. Let us define constants μ = max μ0 : [, ], 3.5) L = + μ )θ t 0 ) 3.6) which are required in Proosition 3.3 and will be used in the sequel.

9 Kh.G. Guseinov, A.S. Nazliinar / J. Math. Anal. Al ) Proosition 3.3. Let >, ε 0,α 0 ) and > where α 0 > 0 is defined by.). Then, there exists ν = ν ε, ) 0, ] such that the inclusion B ) B ) + L ε B ) holds for any, + ν ). Proof. Let where ν ε, ) = min β3 ε, ), β 4 ε, ) β 3 ε, ) = min β 4 ε, ) = min log μ +ε 0 ε log ε ),, ),. It is obvious that ν ε, ) 0, ]. Let, + ν ε, )) and u ) B ) be arbitrarily chosen. We set ut) = u t) u t) μ 0, t [t 0,θ]. 3.7) It can be shown that u ) B ). We denote Aε) = t [t 0,θ]: 0 u t) ε, Bε)= t [t 0,θ]: ε< u t). Let t Aε). Then 0 u t) ε. Since ε<α 0 <, [, ] then we obtain that u t) u t) u t) μ dt εθ t0 ) + μ θ t 0 ) Aε) 0 0 ε ε + μ )θ t 0 ) 3.8) where μ is defined by 3.5). Let t Bε). Then, ε< u t) and consequently ) ) ) μ0 μ0 μ0. 3.9) ε u t) Since, + υ ε, )) then from 3.9) we get ) μ0 u t) ε and hence u t) ) μ0 u t) ε 3.0) for every t Bε). From 3.8) and 3.0) we conclude that

10 356 Kh.G. Guseinov, A.S. Nazliinar / J. Math. Anal. Al ) u ) u ) ε + μ )θ t 0 ) + εθ t 0 ) ε θ t0 )[ + μ ]=ε L where L is defined by 3.6). Since, + ν ε, )) and u ) B ) are arbitrarily chosen, this comletes the roof of the roosition. Proosition 3.4. Let >, ε 0,α 0 ) where α 0 > 0 is defined by.). Then there exists γ = γ ε) 0, ] such that the inclusion B ) B ) + εb ) holds for any, + γ ). The roof of Proosition 3.4 follows from Corollary. and Proosition 3.3. Proositions 3. and 3.4 imly the validity of the following roosition. Proosition 3.5. Let > and ε 0,α 0 ) where α 0 > 0 is defined by.). Then there exists δ = δ ε) 0, ] such that the inequality h B ), B ) ) ε holds for all, + δ ). Finally, from Proositions.5 and 3.5 we obtain the validity of following theorem, which characterizes continuity of the set-valued ma B ) with resect to where + ). Theorem 3.6. Let > and ε 0,α 0 ) where α 0 > 0 is defined by.). Then there exists δ = δε) > 0 such that the inequality h B ), B ) ) ε holds for all δ, + δ). 4. Attainable sets of control systems Consider the control system the behavior of which is described by the differential equation ẋt) = f t,xt),ut) ), xt 0 ) X 0, 4.) where x R n is the hase state vector of the system, u R m is the control vector, t [t 0,θ] is the time and X 0 R n is a comact set. It is assumed that the right-hand side of the system 4.) satisfies the following conditions: 4.A) the function f ) : [t 0,θ] R n R m R n is continuous; 4.B) for any bounded set D [t 0,θ] R n there exist constants L = L D) > 0, L = L D) > 0 and L 3 = L 3 D) > 0 such that ft,x,u ) ft,x,u ) L + L u ) x x +L 3 u u for any t, x ) D, t, x ) D, u R m and u R m ;

11 Kh.G. Guseinov, A.S. Nazliinar / J. Math. Anal. Al ) C) There exists a constant c>0 such that ft,x,u) c + x ) + u ) for every t,x,u) [t 0,θ] R n R m. The set B ) is called the set of admissible control functions and a function u ) B ) is said to be an admissible control function where the set B ) is defined by.). Let u ) B ). The absolutely continuous function x ) : [t 0,θ] R n which satisfies the equation ẋ t) = ft,x t), u t)) a.e. in [t 0,θ] and the initial condition x t 0 ) = x 0 X 0 is said to be a solution of the system 4.) with initial condition x t 0 ) = x 0, generated by the admissible control function u ). The symbol x ; t 0,x 0,u )) denotes a solution of the system 4.) with initial condition xt 0 ) = x 0, generated by the admissible control function u ). Let us define the sets X t 0,X 0, ) = x ; t 0,x 0,u ) ) : [t 0,θ] R n : x 0 X 0,u ) B ) and X t; t 0,X 0, ) = xt) R n : x ) X t 0,X 0, ) where t [t 0,θ]. The set X t; t 0,X 0, ) is called the attainable set of the system 4.) at the instant of time t. It is clear that the set X t; t 0,X 0, ) consists of all x R n to which the system 4.) can be steered at the instant of time t [t 0,θ]. In general the set X t; t 0,X 0, ) R n is not closed see, e.g., [3]) and it is not difficult to verify that it deends on t, t 0, X 0 and continuously. Other roerties of the attainable set X t; t 0,X 0, ) R n and aroximation methods for its numerical construction have been considered in [7,8,0 3]. In this section we show that the attainable set X t; t 0,X 0, ) R n deends on continuously. The following roosition characterizes boundedness of the attainable sets of the system 4.). Proosition 4.. The inequality xt) ρ + r ) exr ) holds for every, + ), x ) X t 0,X 0, ) and t [t 0,θ] where ρ = max x : x X 0, r = cr 0 + ), r 0 = maxθ t 0,, 4.) c>0 is defined by condition 4.C). The roof of the roosition follows from conditions 4.A) 4.C) and Gronwall inequality. Denote D = t, x) [t 0,θ] R n : x ρ + r ) exr ) 4.3) where ρ and r are defined by 4.). According to Proosition 4. we get that t, xt)) D for every, + ), x ) X t 0,X 0, ) and t [t 0,θ]. Therefore, here and henceforth we will have in mind the cylinder 4.3) as the set D in condition 4.B). Note that the continuity roerty of the set-valued ma B ),, + ), imlies the uniform continuity with resect to t) of the set-valued ma X t; t 0,X 0, ),, + ).

12 358 Kh.G. Guseinov, A.S. Nazliinar / J. Math. Anal. Al ) Theorem 4.. Let >, ε 0,α 0 ) where α 0 > 0 is defined by.). Then there exists ξ = ξε) > 0 such that the inequality h c X t 0,X 0, ), X t 0,X 0, ) ) ε holds for any ξ, + ξ) and consequently for all t [t 0,θ] h X t; t 0,X 0, ), X t; t 0,X 0, ) ) ε. ere h c S, G) denotes the ausdorff distance between the sets S C[t 0,θ]; R n ) and G C[t 0,θ]; R n ) and is defined as h c S, G) = max su x ) S ) d c x ), G, su y ) G ) d c y ), S where d c x ), G) = inf x ) y ) c : y ) G, z ) c = max zt) : t [t 0,θ], C[t 0,θ]; R n ) is the sace of continuous functions x ) : [t 0,θ] R n. Proof. Let a 0 = L r 0 + L r 0, b 0 = L 3 exa 0 ) 4.4) where r 0 is defined by 4.). By virtue of Proosition 3.6 for ε b 0 there exists ξ = ξε) such that the inequality h B ), B ) ) ε b 0 4.5) holds for all ξε), + ξε)). Let us choose an arbitrary ξε), + ξε)) and x ) X t 0,X 0, ). Then there exist x 0 X 0 and u ) B ) such that the equality t xt) = x 0 + ) f τ,xτ),uτ) dτ 4.6) t 0 holds for every t [t 0,θ]. According to 4.5) there exists u ) B ) such that u ) u ) ε b ) Let t x t) = x 0 + f τ,x τ), u τ) ) dτ 4.8) t 0 where t [t 0,θ]. Then x ) X t 0,X 0, ). It follows from 4.B), 4.6) 4.8) that xt) x t) L3 ε b 0 + t t 0 L + L u τ) ) xτ) x τ) dτ 4.9) for all t [t 0,θ]. From 4.4), 4.9) and Gronwall s inequality we obtain that the inequality

13 Kh.G. Guseinov, A.S. Nazliinar / J. Math. Anal. Al ) xt) x t) [ θ ε L 3 ex L + L u τ) ) ] dτ b 0 t 0 = ε b 0 L 3 exa 0 ) = ε 4.0) holds for every t [t 0,θ]. Since ξε), + ξε)) and x ) X t 0,X 0, ) are arbitrarily chosen, we get from 4.0) that X t 0,X 0, ) X t 0,X 0, ) + εb c 4.) holds for every ξε), + ξε)) where B c is unique ball of the sace C[t 0,θ]; R n ). Analogously, it is ossible to rove that X t 0,X 0, ) X t 0,X 0, ) + εb c 4.) for every ξε), + ξε)). Thus, inclusions 4.) and 4.) imly the validity of the theorem. References [] J.P. Aubin,. Frankowska, Set Valued Analysis, Birkhäuser, Boston, 990. [] D. Burago, Yu. Burago, S. Ivanov, A Course in Metric Geometry, Grad. Stud. Math., vol. 33, American Mathematical Society, Providence, RI, 00. [3] R. Conti, Problemi di controllo e di controllo ottimale, UTET, Torino, 974. [4] K. Deimling, Multivalued Differential Equations, Walter de Gruyter, New York, 99. [5] A.F. Filiov, Differential Equations with Discontinuous Right-and Sides, Kluwer Acad. Publ., Dordrecht, 988. [6] Sh. u, N.S. Paageorgiou, andbook of Multivalued Analysis, vol., Theory, Kluwer Acad. Publ., Dordrecht, 997. [7] F. Gozzi, P. Loreti, Regularity of the minimum time function and minimum energy roblems: The linear case, SIAM J. Control Otim ) 95. [8] Kh.G. Guseinov, A.N. Moiseev, V.N. Ushakov, On the aroximation of reachable domains of control systems, J. Al. Math. Mech. 6 ) 998) [9] Kh.G. Guseinov, O. Ozer, S.A. Duzce, On differential inclusions with rescribed attainable sets, J. Math. Anal. Al. 77 ) 003) [0] Kh.G. Guseinov, O. Ozer, E. Akyar, V.N. Ushakov, The aroximation of reachable sets of control systems with integral constraint on controls, NoDEA Nonlinear Differential Equations Al. 007), in ress. [].W. Lou, On the attainable sets of control systems with -integrable controls, J. Otim. Theory Al. 3 ) 004) [] M. Motta, C. Sartori, Minimum time with bounded energy, minimum energy with bounded time, SIAM J. Control Otim ) [3] A.I. Panasyuk, Equations of attainable set dynamics, Part : Integral Funnel equations, J. Otim. Theory Al. 64 ) 990)