CARUSO Abstract We study a control problem in a space of continuously di64256eren tiable function when the process is described by an hyperbo lic partial diffe rential equation The complete controllability is obtaine d by the Darbo 64257xed point the ID: 23042 Download Pdf

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CARUSO Abstract We study a control problem in a space of continuously di64256eren tiable function when the process is described by an hyperbo lic partial diffe rential equation The complete controllability is obtaine d by the Darbo 64257xed point the

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ON THE COMPLETE CONTROLLABILITY OF A DISTRIBUTED PARAMETER NONLINEAR CONTROL SYSTEM WITH PARTIALLY COEFFICIENTS A.CARUSO Abstract. We study a control problem in a space of continuously diﬀeren tiable function, when the process is described by an hyperbo lic partial diffe- rential equation. The complete controllability is obtaine d by the Darbo ﬁxed point theorem. Contents 1. Preliminaries 1.1. Distributed Parameter Control Processes of Hyperbolic Type 1 1.2. The Riemann function 1.3. The Darbo Fixed Point Theorem 2. Complete Controllability References 10 1.

Preliminaries 1.1. Distributed Parameter Control Processes of Hyperbolic Type. Con- sider the nonlinear control system of the following type (1.1) xy x,y x,y x,y x,y x,y )+ x,y,z,z ,z ,w x,y = [0 ,a [0 ,b ] ( a,b> 0) where x,y is a state vector, x,y is a control function, x,y , B x,y , C x,y , D x,y and x,y,z,x ,y ,w are respectively matrix-valued and vector-valued functions. In what follows we suppose to ﬁx A,B,C R, ) such that also ,B exist and belong to R, , D R, ) and It is known that if ([0 ,a , ([0 ,b ) are two continuously

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A.CARUSO diﬀerentiable functions,

say ) and for which (0) = (0) = then ﬁxed R, ) there exists only one solution x,y ) of (1.1), continuous with its partial derivatives , z , z xy in R, satisfying the conditions x, 0) = , z (0 ,y ) = [0 ,a [0 ,b Moreover such a solution admits an implicit integral representation given by x,y ) = (0 0; x,y (1.2) u, 0; x,y ) + u, 0) du (0 ,v x,y ) + (0 ,v dv u,v x,y u,v u,v ) + u,v,z u,v ,z u,v ,z u,v ,w u,v )) dudv x,y R, where u,v,x,y u,v x,y is the Riemann function, contin- uous in with its partial derivatives uv xy Notation. Set φ, ) : ([0 ,a , ([0 ,b , (0) = (0) Deﬁnition

1. We say that the distributed parameter nonlinear control system (1.1) is completely controllable if for each ( φ, ∈D and for each there exist R, ) such that the solution of equation (1.1) satisﬁes the condition a,b ) = Notation. For a matrix-valued (or vector-valued) function ) = ( ij )) with bounded entries deﬁned on a space S, the position = max sup ij : 1 p, deﬁnes a norm usually called the sup norm . Moreover the symbol will denote the transpose matrix of the matrix M. Obviously and if ) = jh )) is another matrix-valued bounded functions, it results MN The

following position will be useful in section (2). Notation. (1.3) x,y ) = u,v x,y u,v u,v u,v x,y dudv x,y R.

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ON THE COMPLETE CONTROLLABILITY OF A NONLINEAR HYPERBOLIC SYSTEM 3 1.2. The Riemann function. We now recall some facts, useful in the sequel, about the before mentioned properties of the Riemann function It is known that ﬁxed ( x,y arbitrarily, we can consider the function of the unknown ( u,v to be the only solution of the problem (1.4) uv u,v ) + u,v ) + u,v ) + u,v ) = 0 u,v u,y ) + u,y ) = 0 a,b x,v ) + x,v ) = 0 c,d x,y ) = To emphasize the dependence on

the point ( x,y previous ﬁxed, we can write u,v ) = u,v x,y Starting from (1.4), it can be proved that the function is the only limit solution in the topology of R, ) of the following recursive integral equation +1) u,v x,y ) = x,y ) + x,y (1.5) s,t s,t x,y ) + s,t s,t x,y )+ s,t s,t x,y dsdt u,v R, where we can choose (1) u,v x,y ) = x,y ) + x,y and are, in their turn, the only limit solutions in the topology of ([0 ,a ) and ([0 ,b ) respectively, of the following recursive integral equations +1) x,y ) = s,y s,y ds (1.6) +1) x,y ) = x,v x,s ds [0 ,a [0 ,b with (1) (1) I. So the

function u,v x,y u,v ) is continuous in with its partial derivatives x,y u,v x,y u,v uv x,y u,v To prove that the function can be treated as a useful function of the variable ( u,v,x,y R, we ﬁx ( x,y arbitrarily and, arguing like before, consider the only solution

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A.CARUSO i.e. the above called Riemann function, of the following problem (1.7) uv u,v u,v u,v ) + u,v ) = 0 u,v u,y u,y ) = 0 a,b x,v x,v ) = 0 c,d x,y ) = So also the function u,v x,y u,v ) is continuous in with its par- tial derivatives x,y u,v x,y u,v uv x,y u,v ). At last, by standard technics (see [3],

Chapter 7, N.2), it results that (1.8) u,v x,y ) = x,y u,v u,v x,y R. This relation shows the properties mentioned for after formula (1.2) (for more details refer to [3] Chapter 6 and 7 for the scalar case, [4] for the vectorial cas e). We conclude this section observing that if we suppose that there exists c> 0 such that c, then using (1.4), ..., (1.8) it is easy to verify that ... δ, uv xy hδ, where (...) denotes any one of the ﬁrst order derivatives, exp(12 nc , l = max , h = max ,nc and (1) 1.3. The Darbo Fixed Point Theorem. In Section (2) we obtain complete con-

trollability by using the Darbo ﬁxed point theorem. So we need to recall some fa cts and deﬁnitions. Deﬁnition 2. Let ( S,d ) be a metric space. To each bounded set we associate the following number: ) = inf > 0 : can be covered by a ﬁnite number of balls with radius The function is called the Sadovskii measure of noncompactness. This measure has many important properties (see [1], Chapter 2). Starting f rom these, it is possible to give a general deﬁnition of measure of noncompactness (see [1], Chapter 3). The most known application of a measure of

noncompactness is the following theorem (see [1] Chapter 5): Theorem (Darbo Fixed Point Theorem) Let a bounded closed convex set of a Banach space , and a continuous mapping. Suppose that there exists k< such that )) k for any subset K. Then has a ﬁxed point in K.

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ON THE COMPLETE CONTROLLABILITY OF A NONLINEAR HYPERBOLIC SYSTEM 5 We will apply this theorem to a suitable mapping deﬁned on a given subset xy where the component spaces are deﬁned by the following positions xy R, ) such that ,z ,z xy R, and R, With respect to the sup norm each of these spaces

is a Banach space. In order to apply the Darbo Fixed Point Theorem, we need a characterization of the noncompactness measure that will enable us to construct the requested measure on the space xy . It is known that if ( S,d ) is a compact metric space, then the modulus of continuity of a function S, ) allows us to obtain the requested characterization in the Banach space S, ) endowed with the sup norm kk . More precisely, for each S, ) the function u, ) = sup s,t S, d s,t is called modulus of continuity of the function . If we set B, ) = sup u, ) : and ) = lim B, then (see [1], Chapter

7) ) = In our case (refer to [1], Chapter 8) we can deﬁne a measure of noncompactness on the space xy W, normed with the norm kk xy def. = max kk xy kk by the position ) = max , , , xy , xy W, bounded, where denotes the set of the functions belonging in the projection of the set into W,B denotes the set of the functions belonging in the projection of the set into xy and so on for the remaining sets. 2. Complete Controllability Theorem. Consider the nonlinear control system (1.1) with A,B,C,D,F,G as above. Suppose that A,B,C,F,G do satisfy the following

conditions: (2.1) c, (2.2) x,y,z,x ,y ,w x,y,z,x 00 ,y 00 ,w max 00 00 for some real constants c > k < for each x,y R, x ,x 00 ,y ,y 00 ,z , w moreover suppose that (2.3) det a,b )) = 0

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A.CARUSO Then system (1.1) is completely controllable. Proof. We extend to our case the ideas of [2]. Fix ( φ, ∈D ,z and set = max Deﬁne a continuous mapping xy xy by the position (2.4) z,w ) = ( z,w ,T z,w )) where z,w )( x,y ) = (0 0; x,y (2.5) u, 0; x,y ) + u, 0) du (0 ,v x,y ) + (0 ,v dv u,v x,y u,v z,w )( u,v )+ u,v,z u,v ,z u,v ,z u,v ,T z,w )( u,v )) dudv x,y R, and z,w

) is previously deﬁned as follows z,w )( x,y ) = x,y x,y a,b a,b (0 0; a,b (2.6) u, 0; a,b ) + u, 0) du (0 ,v a,b ) + (0 ,v dv u,v a,b u,v,z u,v ,z u,v ,z u,v ,w u,v )) dudv x,y R. In order to apply the Darbo Fixed Point Theorem (see 1.3), we now deﬁne a convenient bounded closed convex set xy satisfying the hy- pothesis, i.e. and )) k ) for any subset K, where id the measure deﬁned at the end of 1.3. To do this we need to calculate

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ON THE COMPLETE CONTROLLABILITY OF A NONLINEAR HYPERBOLIC SYSTEM 7 z,w x,y , T z,w x,y ) and z,w xy x,y It results: z,w x,y )

= (0 0; x,y u, 0; x,y ) + u, 0) du (2.7) x, 0; x,y ) + x, 0) (0 ,v x,y ) + (0 ,v dv u,v x,y u,v z,w )( u,v )+ u,v,z u,v ,z u,v ,z u,v ,T z,w )( u,v )) dudv x,v x,y x,v z,w )( x,v )+ x,v,z x,v ,z x,v ,z x,v ,T z,w )( x,v )) dv x,y a similar expression holds for z,w x,y ); z,w xy x,y ) = xy (0 0; x,y (2.8) xy u, 0; x,y ) + u, 0) du x, 0; x,y ) + x, 0) xy (0 ,v x,y ) + (0 ,v dv (0 ,y x,y ) + (0 ,y xy u,v x,y u,v z,w )( u,v )+ u,v,z u,v ,z u,v ,z u,v ,T z,w )( u,v )) dudv u,y x,y u,y z,w )( u,y )+ u,y,z u,y ,z u,y ,z u,y ,T z,w )( u,y )) du x,v x,y x,v z,w )( x,v )+ x,v,z x,v ,z x,v ,z x,v ,T z,w

)( x,v )) dv x,y z,w )( x,y ) + x,y,z x,y ,z x,y ,z x,y ,T z,w )( x,y )) x,y R.

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A.CARUSO So, taking into account the results of section (1.2), (1.3) of section (1.1 ), (2.5), ..., (2.8), we can deﬁne z,w xy (2.9) xy where n + 4 l (1 + nc ) + 4 a,b c n + 4 l (1 + cn ) + 4 n (1 + 4 )(1 + cn ) + 2 cl (1 + 2 )(1 + n + 2 (1 + cn )(1 + 6 lh (1 + 1 + 4 lδn (1 + 3 lh It is easy to verify that has the requested properties and that it is mapped by into itself. To verify the last property, ﬁx ∅6 K. For each ( ,y 00 ,y 00 we have z,w )( ,y z,w )( 00 ,y 00 (2.10)

(0 0; ,y (0 0; 00 ,y 00 u, 0; ,y ) + u, 0) du 00 u, 0; 00 ,y 00 ) + u, 0) du (0 ,v ,y ) + (0 ,v dv 00 (0 ,v 00 ,y 00 ) + (0 ,v dv u,v ,y u,v,z u,v ,z u,v ,z u,v ,T z,w )( u,v )) dudv 00 00 u,v 00 ,y 00 u,v,z u,v ,z u,v ,z u,v ,T z,w )( u,v )) dudv so )) = 0 because of the uniform continuity and the uniform boundedness of the functions involved in the right side of (2.10), when ( z,w ) ranges over An analogous consideration holds for ) so that also )) = 0 Moreover, looking at the expression (2.7), one obtains in the same way that (( )) ) =

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ON THE COMPLETE CONTROLLABILITY OF A

NONLINEAR HYPERBOLIC SYSTEM 9 (( )) ) = 0 It remains to estimate from above (( )) xy Writing ex- plicitly the diﬀerence, one observes that, for ( ,y 00 ,y 00 R, the quantity z,w xy ,y z,w xy 00 ,y 00 admits an upper estimates of the kind 00 ,y 00 ,y ,z ,y ,z ,y ,z ,y ,T z,w )( ,y ))+ 00 ,y 00 ,z 00 ,y 00 ,z 00 ,y 00 ,z 00 ,y 00 ,T z,w )( 00 ,y 00 )) where the ﬁrst term of this sum goes to zero when 00 ,y 00 tends to zero, uniformly with respect to ( z,w and the second one is not greater then ,y ,z ,y ,z ,y ,z ,y ,T z,w )( ,y ))+ (2.11) ,y ,z ,y ,z 00 ,y 00 ,z 00 ,y 00 ,T z,w )(

,y )) ,y ,z ,y ,z 00 ,y 00 ,z 00 ,y 00 ,T z,w )( ,y ))+ 00 ,y 00 ,z 00 ,y 00 ,z 00 ,y 00 ,z 00 ,y 00 ,T z,w )( 00 ,y 00 )) Observe now that for any ( ,y arbitrarily ﬁxed, the mapping x,y x,y (2.12) z,w x,y,z x,y ,z ,y ,z ,y ,T z,w )( x,y has the following property: for each > 0 it is possible to ﬁnd a δ > 0 such that ,y implies that x,y lies in the ball with radius centered in ,y ,z ,y ,z ,y ,z ,y ,T z,w )( ,y ; moreover such a does not depend on the point ( ,y ) previously ﬁxed. Indeed this is clear noting that the functions of the set have uniformly bounded

ﬁrst order derivatives and that, as shown before, )) = 0 So the last term in (2.11) does not ex- ceed max ,y 00 ,y 00 ,y 00 ,y 00 the second one goes to zero as 00 ,y 00 tends to zero, uniformly with respect to z,w and, ﬁnally, we have (( )) xy max , All in all we have )) k ) for any subset K. By the Darbo Fixed Point Theorem, there exist ( z, such that ( z, ) = ( z, ) : by formulas (2.5)

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10 A.CARUSO and (2.6) it follows immediately that ( z, ) is the solution we were looking for, according to deﬁnition (1). This concludes the proof. References [1]

J.Banas, K.Goebel Measures of Noncompactness in Banach Spaces Lectures Notes in Pure and Applied Mathematics, Volume 60, [2] C.Dacka On the Controllability of a Class of Nonlinear Systems IEEE Transactions On Au- tomatic Control, Vol. AC-25, N.2, April 1980 [3] H.M.Lieberstein, Theory of Partial Diﬀerential Equation , Mathematics in Science and Engi- neering, Volume 93, Academic Press, Inc. 1972 [4] A.Villani, Un problema al contorno per un sistema lineare iperbolico su un insieme non limi- tato , Le Matematiche, Vol. XXXVI, fasc.II, 1981 Dipartimento di Matematica e

Informatica, Citt a Universitaria, V.le A. Doria 6 - I, 95125 Catania, ITALY E-mail address aocaruso@dmi.unict.it

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