ON THE COMPLETE CONTROLLABILITY OF A DISTRIBUTED PARAMETER NONLINEAR CONTROL SYSTEM WITH PARTIALLY COEFFICIENTS A PDF document - DocSlides

ON THE COMPLETE CONTROLLABILITY OF A DISTRIBUTED PARAMETER NONLINEAR CONTROL SYSTEM WITH PARTIALLY COEFFICIENTS A PDF document - DocSlides

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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 ID: 23042

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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 kk . 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 kk xy def. = max kk xy kk 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.Banas, 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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