IL L MICROCOPY RESOLUTION TEST CHART NATIONAL BUREAU OF STANDARDS- 963-A

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1 AD-A11* 62 WISCONSIN UtIV-NAOIION KATIENATICS RESEARCH CENTER F/6 12/1 SYSTEMS9 POSITIVE AND NESATIVE RESULTS ON -(TC(U) unl nii EC ai N4 SLEM4OO OAA29-80-C-Okl1 F DC RC-S-2310 LN21S~ EN L6I.11

2 IL L MICROCOPY RESOLUTION TEST CHART NATIONAL BUREAU OF STANDARDS- 963-A

3 MRC Technical Summary Report #2310 DISRIBUTMD BILINEAR SYSTEKS: POSITIVE AND NEGATIVE RESULTS ON CONTROLLABILITY M. Slearod Mathematics Research Center University of Wisconsin- Madison 610 Walnut Street Madison, Wisconsin December 1981 CD C-:) (Received November 23, 1981) Approved for public eleasse Distribution DTlOA EECTS E U. S. Army Research Office and Air Force office P. 0. Box of Scientific Research Research Triangle Park Washington, DC North Carolina

4 UNIVERSITY OF WISCONSIN - MADISON MATHEMATICS RESEARCH CENTER DISTRIBUTED BILINEAR SYSTEMS: POSITIVE AND NEGATIVE RESULTS ON CONTROLLABILITY M. Slemrod Technical Summary Report #2310 December 1981 ABSTRACT This paper outlines results presented in [21 on the problem of controlling bilinear distributed parameter systems. Specifically we give results showing that one cannot control a bilinear distributed parameter system to a full open neighborhood of an infinite dimensional state space. Nevertheless we do present a result which allows identification of an accessible set of states for a class of whyperbolic" control systems. AMS (MOS) Subject Classification: 93B05, 93C10, 93C20, 93Cm, F Key Words: Distributed control system, bilinear system, hyperbolic partial differential equations, rod equation. Work Unit Number 1 - Applied Analysis Dept. of Mathematical Sciences, Rensselaer Polytechnic Institute, Troy, NY and the Mathematics Research Center, University of Wisconsin, Madison, WI V Research sponsored in part by the Air Force Office of Scientific Research, Air Force Systems Command, USAF, under Contract/Grant No. AFOSR The United States Government is authorized to produce and distribute reprints for Government purposes not withstanding any copyright notation hereon. Support facilities provided by the United States Army under Contract No. DAAG29-80-C

5 SIGNIFICANCE AND EXPIANTIOM One way to view control of linear partial differential equations where the controls enter as time varying coefficients is as a bilinear distributed system. For example the problem of controlling a rod via the axial load falls in this class. This paper examines the possibility of using such controls to steer the system from one location to another. It shows that this problem is generally ill-posed but may be solved in certain exceptional circumstances. Accession For NTIS GRA&I DTIC TAB Unannounced 0 Justificatto Distribution/ Availability Codes lavail and/or Dist Special ano The responsibility for the wording and views expressed in this descriptive summary lies with NRC, and not with the author of this report.

6 CONTEITS 1. Distributed bilinear control systems 2. The control problem 3. Identification of the accessible set for abstract "hyperbolic" bilinear control systems 4. References 4.

7 DISTRZIBUTED BILINE R SYSTX4Ss POSITIVE AND NEGATIVE RESULTS IN CONTROLLBILITY Me Slearod* 1. Distributed bilinear control systems By a distributed bilinear control system we mean a system of the form ;(t) - Aw(t) + p(t)bwt) 0 (.1) 0(O) = ( 0 6 X, (1.2) where A generates a C 0 semigroup of bounded linear operators on a (possible complex) Banach space X, S : X + X is a bounded linear operator, and p G L 1 ([0,T]; R) is a real valued control defined on a specified ) interval [0,T]. Of particular interest in applications is the abstract "hyperbolic" bilinear control system given by a(t) + Au(t) + p(t)bu(t) m 0, (1.3) u(o) - u 0 Q D(AW, u10 - u 1 6 H 1.4) * where A is a positive definite self-adjoint operator with dense domain D(A) in a real Hilbert space H, B is a bounded linear operator from D(AY2) to H, and p (the control) is again in L 1 ([0,T]; R). We suppose A - 1 is compact and A has simple eigenvalues A2, n m 1,2,..., where 0 < A < A <***. Then there exists a corresponding sequence {*n } of 1 2 eigenfunctions: A# - n*n, <*n,* > - 6 where <0,*> denotes the inner n n n n mn H n H product in H. Dept. of Mathematical Sciences, Rensselaer Polytechnic Institute, Troy, NY and the Mathematics Research Center, University of Wisconsin, Madison, WI Re'search sponsored in pare by the Air Force Office of Scientific Researcf, Air Force Systems Command, USAF, under Contract/Grant No. AFOSR The United States Government is authorized to produce and distribute reprints for Government purposes not withstanding any copyright notation hereon. Support facilities provided by the United States Army under Contract No. DAAG29-80-C

8 While we could rewrite (1.3), (1.4) in first order form (1.1), (1.2) in the usual manner of dynamical systems (see for example [1], [2)) it is preferable for our purposes to introduce a complex structure similar to that used for Hamiltonian systems. Let H denote the complexified illbert space H O ih with inner product defined by <1 + iy 1, x 2 + iy 2 > - <x 1, x 2 >H + <yl,y 2 > H + i[<yl'x 2 >H - <xly2>. ] for x 1,x 2,yly 2 Q H. Let z(t) - 1/2u(t) + iu(t). Then (1.3), (1.4) becomes ;(t) -Az(t) + p(t)9z(t), (1.5) z(0) UA 12u / +i U G1H (1.6) where A -) -il/2, 8 - -im /2Re, and thus (1.3), (1.4) has been put in the form (1.1), (1.2) with X -H. Example. The rod equation with hinged ends. Consider the system with boundary conditions with initial conditions utt + Uxxxx + p(t)uxx -, 0 < x < 1 (1.7) u = Uxx M 0 at x-0,1, (1.8) u(x,o) - u 0 (x), ut(x,0) - u (X), 0 < x < 1 (1.9) In the notation of (1.3), (1.4) we have d4 d2 A d A 4 B B. d -2, H2 H - L 2 (0,1), D(A) - {u Q H4(0#1); 4 dx dx UP u 69 o )) D0,12. ) - H(0,1)n M(0,1), n nx #n = r sin nix, n - 1,2,... Here p(t) represents the axial load on the rod. -2-

9 2. The control problem Consider the system 1.1), (1.2). The controllability poblem is (P) Given h 0 X, find p 6 L ([O,T]! R) so that the (generalized) solution of 1.1), (1.2) with control denoted by w(tsp, 0 ) satisfies (.o(t;p,w O 0 ) h. We note two important features of (P). First we observe that even though 1.1) is a linear evolution equation in w for fixed p the map wl(tg,w 0 is in fact a nonlinear function of p : LI (0,T] R) + X. Secondly (and of great importance for distributed systems) we see that (P) in fact asks us to control a (generally) infinite dimensional system with controls p(t) in onedimensional real space for each t. The first observation means that our analysis may likely resort to local theory (the inverse function theorem). The second observation is more serious since it will generally imply the map i (T;6,W 0 ) : Lr ([O,T]; R) + C([O,T]; X) is compact for r > 1. More precisely it was proved in [2]. i ' Theorem 1. Let X be a Banach space with dim X =.If a > T > 0 and Pn + p weakly in LI((o,T]; R) then w(*p n, W O ) + w(*;p,( O ) strongly in C([0,T]! X). Moreover the set of states accessible from w 0 defined by S(w O ) U w(tjp,t 0 ) 0 t)0 r>1 is contained in a countable union of compact sets of X, and in particular has dense complement. The proof of Theorem I is quite technical and is given in [2). Its importance in controllability of (1.1), (1.2) is readily seen, however. For if the set of states accessible from w 0 S(b 0 ) has dense complement we shall -3-

10 never be able to steer to an open neighborhood in X of W o. In other words except for an exceptional set of h 9 X (P) is ill-posed. 3. Identification of the accessible set for abstract "hyperbolic" bilinear control systems. Having noted in Meorem 1 the inability to control (1.1), (1.2) to any open neighborhood of X, we turn instead to trying to identify what states are accessible from a given w 0 9 X. Specifically we consider the abstract "hyperbolic" bilinear control system (1.3), (1.4). As the basis {(n) of H may be regarded as a basis of H, any z r H may be expanded as a Fourier series in the basis {#n. hence we may write an s(t) -I z -n(t)#n (3.1) n-1 We assume in addition that <B~nm> H - 0 for n j m (3.2) and assume <B*,,tn> -bn O 0 Then substitution of (3.1) into (1.5) yields the infinite system of ordinary differential equations with initial conditions b n (t) - int) - i(t)z Re zn(t), n - 1,2,... (3.3) n z n(0) = zn zon <Z(O), n > (4) In itself (3.3) is not much of an irprovement over the earlier formulations of our problem in that the map p + (z(t;pz(0))} from Lt((0,T]; R) into (the natural state space for (33), (3.4)) C(I0,T] 1 ) is still compact (now for r 0 1). However (3.3) allows us to make the remarkable change of variables -4-

11 A:) zn(t) b : n~t - [- " exp il,t + n (t)) -1] 35 n n where P(t) - p(s)ds equations A straightforward computation shows that ( n(t)} satisfies the n p)z~ b b nlt) - i 2t) n ( n(t) + 1)expf2i(Xnt +.A-P(t))], (3.6) On n n where we assume zon k 0. n (0) - 0, n 1,2,..., (3.7) 3 It is not hard to show (3.6), (3.7) has a solution (Cn(t;p)l 6 C((0,T); 12) which is C 1 in p as a map from L 2 ((0,T]; R) to 20 The amazing thing is that this map is not compact. Hence we may attempt to control (3.6) in a neighborhood in I of the initial state (0W. The 2 natural way to do this is, as remarked earlier, to apply the inverse function theorem and show Dp (n (T;O)) (the Frechet derivative of (Cn(T;p)} with respect to p evaluated at p = 0) is an isomorphism from L 2 ((0,TJ; R) to 2" Actually this won't be the case since D {C (T;0)) isn't one to one but 2~p n it is onto. Fortunately the "local onto theorem" (3) will still imply the non-unique solvability of C (T;p) = h n, n = 1,2,3,... for some T > 0 and p G L 2 ((0,TJ, R) when Nth n is sufficiently small. Now knowing the accessible states of (3.6), (3.7) is an open neighborhood of (0) in t2 we can translate back via (3.5) to find the accessible states of (3.3) from data (3.4). For example for the rod equation with hinged ends (1.7), (1.8) <B#n'#m> H = 0 and assumptions (3.2) are satisfied. In fact we can prove Theorem 2. Assume zon 1 0 for the rod equation with hinged ends (1.7), 1.8). Then there exists e > 0 so that if ithn}i I < E we can solve 2-5-

12 2 u (.) - (1 + h),on, n- 1,2,... for infinitely many p Q L2(0, j ; I) with f.2s'p(t)dt 0. Thus while we cannot hit all states in a small A 2 neighborhood of (a I we can hit those of the form ( + h )z 1, l(hn)i < e. Of course on n on n &2 using the definition of x this result could be translated into a statement (albeit messy) regarding u and ut for (1.7), (.1.8), (1.9). Furthermore using the above local result we can use a standard argument to prove Corollary. For (1.7), (1.8) with zon.p 0 n = 1,2,... the set of states accessible for {z On is dense in H. -6-

13 REFERECES 1. b all, J. M. and M. Sleearod, Nonharmonic Fourier series and the stabilization of distributed semi-linear control systems, Communications on Pure and Applied Mathematics, 32, , (1979). 2. Ball, J. Mt., M. Slearod and J. E. Marsden, Controllability for distributed bilinear systems, to appear SIAN Journal on Control and Optimization. * 3. Luenberger, D. G., Optimization by Vector Space Methods, John Wiley Sons, New York (1969). MS/j vs/ -7-

14 SECURITY CLASSIFICATION Of THIS PAG (rih on Daita F red) _ REPORT DOCUMENTATION PAGE ri:at) CM111r, ORM 1. REPORT NUMDLR -2. VOV ACCESSION NO 3. RECIPIENT*S CATALOG NUMULR ~~#2310._- 4. TITLE (and ublitle) S". TYPE OF REPORT & PERIOD COVEREO Distributed Bilinear Systems: Positive and Summary Report - no specific Negative Results on Controllability 6. reporting period 6PERFORMING ORG. REPORT NUMBER 7. AUTHOR(.). CONTRACT OR GRANT NUMDER(8) APOSR M. Slemrod DAAG29-80-C S. PERFORMING ORGANIZATION NAME AND ADDRESS 10. PROGRAM ELEMENT. PROJECT. TASK, Mathematics Research Mathematics CneUiest Uni Research"Center s of fwr AREA A WORK ntnme UNIT NUMBERS Work Unit Number Walnut Street Wisconsin Applied Analysis Madison, Wisconsin Il. CONTROLLING OFFICE NAME AND ADORESS 1*. REPORT DATE December 1981 (see Item 18 below) 13. NUMBER OF PAGES 7 $4. MONITORING %GENCY NAME A AOCRESS(I" dillemt from Controlin4 Office) 15. SECURITY CLASS. (1 thie report) _ UNCLASSIFIED M e, OECL. ASSI IC ATION/OWVN GR ADING SCHEDULE 16. DISTRIBUTION STATEMENT (of tl Report) S Approved for public release; distribution unlimited. 17. DISTRIBUTION STATEMENT (of the abtmacil entereodin Block 20f ilditllereat from Report) IS, SUPPLEMENTARY NOTES U. S. Army Research Office and Air Force Office P. 0. Box of Scientific Research Research Triangle Park Washington, DC North Carolina ). KEY WORDS (Continue on rever** aide if neceemy And idently by block number) Distributed control system, bilinear system, hyperbolic partial differential gquations, rod equation. 20. A83 4 ACT (Continue m reverse sid It nceseay and identify by Moc f let) This paper outlines results presented in712] on the problem of controlling bilinear distributed parameter systems. Specifically we give results showing that.one cannot control a bilinear distributed parameer system to a full open neighborhood of an infinite dimensional state space. Nevertheless we do present a result which allows identification of an accessible set of states for a class of "hyperbolic" control systems. DD 1473 EITION o,, I NOVS is s O.SOLTe UNCLASSIFIED.ECURITY CLASSIFICATION OF THIS PAGIE (B1ito De Eftfoerd)

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