Two-point Output Feedback Boundary Control for Semilinear Hyperbolic Systems ⁎

11th IFAC Symposium on Nonlinear Control Systems 11th IFAC Symposium on Nonlinear Control Systems Vienna, Austria, Sept. 4-6, 2019 11th IFAC Symposium on Nonlinear Control Systems Available online at www.sciencedirect.com Vienna, Austria, Sept. 4-6, 2019 11th IFAC Symposium on Nonlinear Control Systems Vienna, Austria, Sept. 4-6, 2019 Vienna, Austria, Sept. 4-6, 2019

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IFAC PapersOnLine 52-16 (2019) 54–59

Two-point Output Feedback Boundary Two-point Output Feedback Boundary Two-point Output Feedback Boundary Control for Semilinear Hyperbolic Two-point Output Feedback Boundary Control for Semilinear Hyperbolic Control for Semilinear Hyperbolic   Hyperbolic Systems Control for Semilinear Systems Systems Systems  Maksim Dolgopolik ∗,∗∗∗ Alexander L. Fradkov ∗∗∗

Maksim Dolgopolik ∗,∗∗∗ L. Fradkov ∗∗∗ ∗,∗∗,∗∗∗ ∗,∗∗∗ Alexander Boris Andrievsky Maksim Dolgopolik L. Fradkov ∗∗∗ ∗,∗∗,∗∗∗ ∗,∗∗∗ Alexander ∗∗∗ Boris Andrievsky Maksim Dolgopolik Alexander ∗,∗∗,∗∗∗ Boris Andrievsky ∗,∗∗,∗∗∗L. Fradkov Boris Andrievsky ∗ ∗ Institute of Problems in Mechanical Engineering, Russian Academy ∗ Institute of Problems in Mechanical Engineering, Russian Academy of Sciences, Saint Petersburg, Russia Mechanical Engineering, Russian Academy ∗ Institute of of Problems Sciences, in Saint Petersburg, Russia (e-mails: (e-mails: Institute of Mechanical Engineering, Russian Academy of Problems Sciences, in Saint Petersburg, Russia (e-mails: [email protected], [email protected], [email protected], [email protected], of Sciences, Saint Petersburg, Russia (e-mails: [email protected]). [email protected], [email protected], [email protected]). ∗∗ [email protected], [email protected], State [email protected]). ∗∗ Saint Petersburg Saint Petersburg State University, University, Saint Saint Petersburg, Petersburg, Russia Russia ∗∗∗ [email protected]). ∗∗ University ITMO, Saint Petersburg, Russia Saint Petersburg State University, Saint Petersburg, Russia ∗∗∗ ∗∗ University ITMO, Saint Petersburg, Russia ∗∗∗ Saint Petersburg State University, Saint Petersburg, University ITMO, Saint Petersburg, Russia Russia ∗∗∗ University ITMO, Saint Petersburg, Russia Abstract: A new control Abstract: A new control problem problem is is posed posed and and solved: solved: regulation regulation problem problem for for the the oneonedimensional A Klein-Gordon and semilinear wave and equations with Neumannproblem boundary conditions Abstract: new controland problem is posed solved: regulation forconditions the onedimensional Klein-Gordon semilinear wave equations with Neumann boundary Abstract: A new control problem is posed and solved: regulation problem for the onein the the case case when when the control control acts at both both endsequations of the the space space interval (“two-point (“two-point control”). dimensional Klein-Gordon and acts semilinear wave with Neumann boundary conditions in the at ends of interval control”). dimensional Klein-Gordon and semilinear wave equations with Neumann boundary conditions A control control algorithm based on the the speed-gradient method is proposed. proposed. The global exponential exponential in the casealgorithm when thebased control actsspeed-gradient at both ends method of the space interval The (“two-point control”). A on is global in the case when thebased control actsspeed-gradient at the bothcase ends of the space interval (“two-point control”). A control algorithm on the method is proposed. The global exponential stability of the closed loop system for of the Klein-Gordon equation is established by stability the closedbased loop on system for the case of method the Klein-Gordon equation is established by ofalgorithm A control the speed-gradient is proposed. The global exponential means of a new Lyapunov functional. This results is extended to the case of the semilinear wave stability of the closed loop system for the case of the Klein-Gordon equation is established by means of of Lyapunov a new This results extended to the case of theissemilinear wave stability the closed loopfunctional. system for the case of is Klein-Gordon equation established by equation by means of linearization. linearization. The two-point energy control problem for the sine-Gordon means of by a new Lyapunov functional. This results isthe extended to the case of for the the semilinear wave equation means of The two-point energy control problem sine-Gordon means of a new Lyapunov functional. This results is extended to the case of the semilinear wave equation by means of linearization. The two-point energy control problem for the sine-Gordon and semilinear semilinear wave wave equations equations is is analyzed analyzed by by simulation. simulation. It It is is demonstrated demonstrated that that the the proposed proposed and equation by means ofequations linearization. The two-point energy problem forthat thethe sine-Gordon and semilinear wave is provide analyzed by simulation. Itcontrol is demonstrated proposed two-point control algorithm may 30% faster transients. two-point algorithm may faster transients. control and semilinear wave equations is provide analyzed30% by simulation. It is demonstrated that the proposed two-point control algorithm may provide 30% faster transients. © 2019, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. two-point control algorithm may provide 30% faster transients. Keywords: distributed-parameter distributed-parameter system, system, boundary boundary control, control, speed-gradient, speed-gradient, Klein-Gordon Klein-Gordon Keywords: equation, semilinear semilinear wave equation, equation, energyboundary control. control, speed-gradient, Klein-Gordon Keywords: distributed-parameter system, equation, wave energy control. Keywords: distributed-parameter system, equation, semilinear wave equation, energyboundary control. control, speed-gradient, Klein-Gordon equation, semilinear wave equation, energy control. 1. INTRODUCTION INTRODUCTION sor 1. sor and and actuator actuator was was studied. studied. The The wave wave equation equation with with 1. INTRODUCTION sor and actuator boundary was studied. The wave equation with Van der Pol-type conditions was considered in Van and der Pol-type boundary conditions wasequation considered in 1. INTRODUCTION sor actuator was studied. The wave with (Feng, 2016). Boundary stabilization problem for general Van der Pol-type boundary conditions was considered in Many control methods for distributed parameter systems Many control methods for distributed parameter systems (Feng, 2016). Boundary stabilization problem for general Van der Pol-type boundary conditions was considered in Many control methods forindistributed parameterincluding systems (Feng, semilinear wave equations was considered in (Kobayashi, 2016). Boundary stabilization problem for general (PDE) were developed the past decades, (PDE) were developed indistributed the past decades, including semilinear wave equations was considered in (Kobayashi, (Feng, 2016). Boundary stabilization problem for general Many control methods for parameter systems semilinear wave equations was considered in (Kobayashi, (PDE) were developed the past and Terushkin, 2016), the the backstepping backstepping methodin(Krstic (Krstic and Smyshlyaev, Smyshlyaev, 2008), 2003a; decades, including the method and 2008), 2003a; Fridman Fridman and Terushkin, 2016), while whileinin in(Kobayashi, the particparticwave equations wasequation considered (PDE) were developed in(Krstic the past decades, including the backstepping method and Smyshlyaev, 2008), semilinear ular case of the sine-Gordon it was analyzed in 2003a; Fridman and Terushkin, 2016), while in the particoptimal control (Tr¨ o ltzsch, 2010; Lasiecka and Triggiani, ular case of the sine-Gordon equation it was analyzed in optimal control (Tr¨ oltzsch,(Krstic 2010; and Lasiecka and Triggiani, 2003a; Fridman and Terushkin, 2016), while in the particthe backstepping method Smyshlyaev, 2008), optimal control (Tr¨ oltzsch, 2010; Lasiecka and Triggiani, (Kobayashi, 2003b, 2004). Exact controllability of semilinular case of the sine-Gordon equation it was analyzed in 2000), robust control (Christofides, 2001), adaptive control 2000), robust control (Christofides, 2001), adaptive control (Kobayashi, 2003b, 2004). Exact controllability of semilinular case equation of 2003b, the sine-Gordon equation it was analyzed in optimal control (Tr¨ oltzsch, 2010; H Lasiecka and(Orlov Triggiani, 2000), robust control (Christofides, 2001), adaptive control ear wave was analyzed in (Coron, 2007; Zuazua, (Kobayashi, 2004). Exact controllability of semilin-control and (Smyshlyaev and Krstic, 2010), ∞ (Smyshlyaev and Krstic, 2010), H2001), wave equation was analyzed in (Coron, 2007; Zuazua, (Orlov and ear ∞ -control (Kobayashi, 2003b, 2004). Exact controllability of semilin2000), robust control (Christofides, adaptive control ear wave equation was analyzed in (Coron, 2007; Zuazua, -control (Orlov and (Smyshlyaev and Krstic, 2010), H 1993). However, to the best knowledge, there Aguilar, 2014), etc. Among numerous other problems, ∞ 1993). However, towas the analyzed best of of authors’ authors’ knowledge, there Aguilar, 2014), Among numerous other(Orlov problems, wave equation in (Coron, 2007; Zuazua, and ear (Smyshlyaev andetc. Krstic, 2010), H∞ -control Aguilar, 2014), etc. Among numerous othera semilinear problems, are no existing results on the regulation problem for the 1993). However, to the best of authors’ knowledge, there boundary control of the wave equation and are no existing results on the regulation problem for the boundary2014), controletc. of the wave numerous equation and a semilinear 1993). However, to the best of authors’ knowledge, there Aguilar, Among other problems, wave or semilinear wave equation with Neumann boundary are no existing results on the regulation problem for the wave equation (e.g. the sine-Gordon equation) have atboundary control of wave equation and a semilinear wave or semilinear wave equation with Neumann boundary wave equation (e.g. the sine-Gordon equation) have atare noorexisting results on thethe regulation problem for the boundary control ofattention the sine-Gordon waveofequation and aInsemilinear conditions in the case when the control acts at the wave semilinear wave equation with Neumann boundary tracted significant researchers. (Zuazua, wave equation (e.g. the equation) have atconditions in the case when thewith the Neumann control acts at the tracted significant attention of researchers. In (Zuazua, wave or semilinear wave equation boundary wave equation (e.g. the sine-Gordon equation) have atconditions in the case when the the control acts at the tracted significant attention of researchers. In (Zuazua, whole of the (at ends the 1990; Lasiecka and Triggiani, 1992) uniform whole boundary boundary the domain domain (at both both ends of of acts the interval interval 1990; Lasiecka and attention Triggiani, of 1992) uniform stabilization stabilization in theof when At the theglance control atseem the tracted significant researchers. In boundary (Zuazua, conditions in the one-dimensional case). it might whole boundary of case the domain (atfirst both ends of the interval of the wave equation via linear and nonlinear 1990; Lasiecka and Triggiani, 1992) uniform stabilization in the one-dimensional case). At first glance it might seem of the wave equation via linear and nonlinear boundary whole boundary of the domain (at both ends of the interval 1990; Lasiecka and Triggiani, 1992) uniform stabilization that problem ones thethis one-dimensional case).simpler At first than glancethe it might seem feedback was studied. for the of the wave via regulation linear and problem nonlinear that this problem is is much much simpler than the ones when when feedback was equation studied. The The problem forboundary the wave wave in in the one-dimensional case). At of first glance it might seem of the wave equation via regulation linear and nonlinear boundary the control acts only on a part the boundary or that this problem is much simpler than the ones when equation with disturbance or time delay was analyzed in feedback was studied. The regulation problem for the wave control acts only on a part of the boundary or when when equation withstudied. disturbance or time delay was for analyzed in the that this problem is much simpler than the ones feedback was The regulation problem the wave the control acts only on a part of the boundary or when equation with disturbance or time delay was analyzed in are non-collocated with control. (Guo Zhou, 2015; and Guo, 2016) and the observations observations are withboundary control. However, However, (Guo and andwith Zhou, 2015; Feng Feng and Guo, 2016) and (Frid(Fridcontrol actstrue, onlynon-collocated on astandard part of approaches the or when equation disturbance oral., time delay was analyzed in the this is far from and to stability the observations are non-collocated with control. man et al., 2010; Wang et 2011) respectively, while (Guo and Zhou, 2015; Feng and Guo, 2016) and (Fridthis is far from true, and standard approaches toHowever, stability man et al., 2010; Wang et al., 2011) respectively, while the observations are non-collocated with control. However, (Guo and Zhou, 2015; Feng and Guo, 2016) and (Fridanalysis in is farfail from true,case. and standard approaches to stability the tracking problem for this equation was in man et al., 2010; Wang 2011) respectively, while analysis in this this the tracking problem for et thisal., equation was considered considered in this this is farfail from true,case. and standard approaches to stability man et et al., 2010; Wang et al., 2011) respectively, while analysis fail in this case. (Sagert al., 2013; Guo and Guo, 2016). A number of pubthe tracking problem for this equation was considered in (Sagert et al.,problem 2013; Guo and Guo, 2016).was A number of pubIn this paper we study the analysis fail in this case. the tracking for this equation considered in (Sagert et al., 2013; Guo and Guo, 2016). A number of pubIn this paper we study the regulation regulation problem problem for for the the oneonelications been devoted to stabilization of an licationsethas has been devoted toGuo, stabilization ofnumber an anti-stable anti-stable dimensional Klein-Gordon and semilinear wave equations In this paper we study the regulation problem for the one(Sagert al., 2013; Guo and 2016). A of pubdimensional Klein-Gordon and semilinear wave equations wave equation with unknown parameters and disturbances lications has been devoted to stabilization of an anti-stable In this paperKlein-Gordon weboundary study the regulation problem forequations the onewave equation withdevoted unknown parameters and disturbances dimensional and semilinear wave with Neumann in case when lications has been toand stabilization ofFeng an anti-stable with Neumann boundary conditions conditions in the thewave case equations when the the (Guo and Guo, 2013; Guo Jin, 2015; and Guo, wave equation with unknown parameters and disturbances dimensional Klein-Gordon and semilinear with Neumann boundary conditions in the case when the (Guo and Guo, 2013; Guo and Jin, 2015; Feng and Guo, control acts at both ends of the interval. We propose a wave equation with unknown parameters and disturbances control acts at boundary both endsconditions of the interval. We propose a 2017), while in (Guo and Xu, 2007; Guo and Guo, 2009) (Guo and Guo, 2013; Guo and Jin, 2015; Feng and Guo, with Neumann in the case when the 2017), while in (Guo and Xu, 2007; Guo and Guo, 2009) speed-gradient control law for solving this problem, and control acts at both ends of the interval. We propose (Guo and Guo, 2013; Guo and2007; Jin, 2015; FengGuo, and 2009) Guo, 2017), while in the (Guo and Xu, Guo and speed-gradient control law of forthe solving this We problem, anda stabilization of wave equation with noncollocated sencontrol acts at both ends interval. propose a stabilization wave with noncollocated sen- speed-gradient with the new function prove that the law for solving problem, 2017), while of in the (Guo andequation Xu, 2007; Guo and Guo, 2009) with the use use of of aacontrol new Lyapunov Lyapunov functionthis prove that in in and the stabilization the wave equation with noncollocated sen The stabilityofanalysis speed-gradient control law for solving this problem, and of the closed-loop system (Sect. 3) was with the use of a new Lyapunov function prove that in the case of the Klein-Gordon equation the solution of  The stabilityofanalysis stabilization the wave equation with system noncollocated of the closed-loop (Sect. 3) senwas case equation the zero zero solution of the the  with of thethe useKlein-Gordon of a new Lyapunov function prove that in performed in IPME RAS and supported by thesystem President of Russian The stability analysis of the closed-loop (Sect. 3) was case of the Klein-Gordon equation the zero solution of the closed-loop system is globally exponentially stable. Then performed in IPME RAS and supported by the President of Russian  closed-loop system is globally exponentially stable. of Then The stability analysis of the closed-loop system (Sect. 3) was Federation grant forRAS the and support of young Russian scientist (grant performed in IPME supported by the President of Russian case of the Klein-Gordon equation the zero solution the we apply linearization technique to (locally) extend this closed-loop system is globally exponentially stable. Then Federation grant for the support of young Russian scientist (grant performed in IPME RAS and supported President Russian we apply linearization technique to (locally)stable. extendThen this number MK-3621.2019.1). The rest of by thethe results wereofobtained closed-loop system is globally exponentially Federation grant for the support of young Russian scientist (grant result to case semilinear wave equation. number MK-3621.2019.1). The rest of theRussian results scientist were obtained we apply linearization (locally) extend this Federation grant for the support of young (grant under theMK-3621.2019.1). support of RFBR (grant 17-08-01728) and the Government result to the the case of of the thetechnique semilinearto equation. number The rest of the results were obtained we apply linearization technique towave (locally) extend this under theMK-3621.2019.1). support of RFBR (grant 17-08-01728) and the Government result to the case of the semilinear wave equation. number The rest of the results were obtained of Russian Federation (Grant 08-08). under the support of RFBR (grant 17-08-01728) and the Government result to the case of the semilinear wave equation. of Russian Federation (Grant 08-08). under the support of RFBR (grant 17-08-01728) and the Government

of Russian Federation (Grant 08-08). of Russian © Federation (Grant 08-08). Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. 2405-8963 2019, IFAC (International Copyright © 2019 IFAC 56 Copyright 2019 IFAC 56 Control. Peer review© under responsibility of International Federation of Automatic Copyright © 2019 IFAC 56 10.1016/j.ifacol.2019.11.755 Copyright © 2019 IFAC 56

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Maksim Dolgopolik et al. / IFAC PapersOnLine 52-16 (2019) 54–59

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3. STABILITY ANALYSIS

Apart from the regulation problem, we also discuss the more general energy control problem. Energy control problem for the sine-Gordon and semilinear wave equation was considered in (Dolgopolik et al., 2016; Orlov et al., 2017; Dolgopolik et al., 2018; Orlov et al., 2018). However, similar to regulation problem, this problem has never been considered in the case when the control acts on the whole boundary of the domain. We design a speedgradient control law for solving this problem, and study its performance via numerical simulation.

Now we can turn to the stability analysis of the closedloop system (5)–(7). At first, let us consider the linearized system (8) ztt (t, x) − kzxx (t, x) + βz(t, x) = 0, (9) z(0, x) = z0 (x), zt (0, x) = z1 (x), zx (t, 0) = γ1 zt (t, 0), zx (t, 0) = −γ2 zt (t, 1). (10) where β = f  (0) (recall that f (0) = 0). Denote by   1 1 2 H0 (z) = zt + kzx2 + βz 2 dx 2 0 the Hamiltonian for the linearized equation. We need the following simple auxiliary result, which follows directly from the fundamental theorem of calculus and the inequality (a + b)2 ≤ 2a2 + 2b2 . Lemma 1. For any z ∈ H 1 (0, 1) one has  1  1  2  2 2 z dx ≤ 2 min z (0), z (1) + 2 zx2 dx, (11) 0 0  1 z 2 (0) ≤ 2z 2 (1) + 2 zx2 dx. (12)

2. PROBLEM FORMULATION AND OUTPUT FEEDBACK DESIGN In the sequel, we study the following initial-boundary value problem for the semilinear wave equation with Neumann boundary conditions:   (1) ztt (t, x) − kzxx (t, x) + f z(t, x) = 0, z(0, x) = z0 (x), zt (0, x) = z1 (x), (2) (3) zx (t, 0) = u1 (t), zx (t, 1) = u2 (t),   y(t) = zt (t, 0), zt (t, 1) . (4) Here x ∈ [0, 1], t ≥ 0, the parameter k > 0, and the functions z0 , z1 : [0, 1] → R are given, u(t) = (u1 (t), u2 (t)) is a control input, y(t) is the output (boundary measurements), and f : R → R is a continuously differentiable function such that f (0) = 0. Our aim is to design an output feedback control law that makes the zero solution of (1)–(4) asymptotically stable.

0

Theorem 2. Suppose that β > 0. Then for any γ1 ≥ 0 and γ2 > 0 there exist M > 0 and δ > 0 (independent of z0 and z1 ) such that H0 (z(t)) ≤ M H0 (z(0))e−δt for any t ≥ 0, where z(t, x) is a solution of (8)–(10).

Proof. Denote H0 (t) = H0 (z(t, x)). Differentiating H0 (t), applying (8) and (10), and integrating by parts one obtains that d H0 (t) = −γ1 kzt (t, 0)2 − γ2 kzt (t, 1)2 ≤ 0. (13) dt Introduce the Lyapunov function  1   1 xzt zx dx + c zzt dx V (t) = H0 (t)+ε 0 0 (14)  ck  2 2 γ2 z (t, 1) + γ1 z (t, 0) , +ε 2 where ε > 0 and c ∈ (0, 0.5). Observe that  1      1 1 2 1 1 2   xz z dx ≤ z dx + z dx t x t   2 2 0 x 0 0   1 H0 (t). ≤ max 1, k Similarly, one has   1     1 1   , H0 (t). zz dx ≤ max t   k β 0 Therefore for any ε ∈ (0, 1/η) one has 0 ≤ (1 − εη)H0 (t) ≤ V (t)  (15) ck  2 γ2 z (t, 1) + γ1 z 2 (t, 0) , ≤ (1 + εη)H0 (t) + ε 2 where η = max{1, 1/k}+max{c/k, c/β}. Applying (8) and integrating by parts one obtains that   1 d 1 xzt zx dx = x(kzxx − βz)zx dx dt 0 0  1  k 1 2 + xzt ztx dx = − z dx + kzx2 (t, 1) 2 0 x 0   β 1 2 β 2 1 1 2 + z dx − z (t, 1) − z dx + zt2 (t, 1), 2 0 2 2 0 t

Let us apply the speed-gradient algorithm (see e.g. (Fradkov et al., 1999; Fradkov, 2007)). To this end, introduce the  1 goal function Q(z(t)) = 21 0 zt2 + kzx2 dx, where z(t) is the profile of z at a certain t ≥ 0, i.e. (z(t))(x) = z(t, x) for all x ∈ [0, 1]. Formally differentiating this function along solutions of (1)–(4), and integrating by parts one obtains that d Q(z(t)) dt  1   zt f (z) dx + k zx (t, 1)zt (t, 1) − zx (t, 0)zt (t, 0) =− 0  1 =− zt f (z) dx + ku2 (t)y2 (t) − ku1 (t)y1 (t). 0

Then according to the speed-gradient algorithm one de d fines u = −Γ∇u dt Q(z(t)) , where Γ is a positive definite gain matrix. Therefore, we can define u1 (t) = γ1 y1 (t) = γ1 zt (t, 0), u2 (t) = −γ2 y2 (t) = −γ2 zt (t, 1), where γ1 , γ2 ≥ 0 are control gains. Then the closed-loop system takes the form   (5) ztt (t, x) − kzxx (t, x) + f z(t, x) = 0, z(0, x) = z0 (x), zt (0, x) = z1 (x), (6) zx (t, 0) = γ1 zt (t, 0), zx (t, 1) = −γ2 zt (t, 1). (7) Under some natural assumptions on the function f the well-posedness of this system can be readily proved with the use of standard existence theorems for Lipschitz perturbations of linear evolution equations (Pazy, 1983, Chapter 6). Below, we prove that the zero solution of this system is exponentially stable.

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and d dt



1

zzt dx = 0



1

zt2 dx +

0

= −k



1 0

zx2



0

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where β = f  (0), and g(t, x, z) = f (z(t, x)) − βz(t, x). By the Sobolev imbedding theorem there exists C > 0 such that for all t ≥ 0 one has

1

z(kzxx − βz) dx  1 1 zt2 dx − β z 2 dx 0

max |z(t, x)| ≤ Cz(t)H 1 (0,1)    2 2 , H0 (z(t)). ≤ C max k β

x∈[0,1]

0

dx + kz(t, 1)zx (t, 1) − kz(t, 0)zx (t, 0).

(16)

Hence and from the fact that f is continuously differentiable it follows that for any η > 0 there exists r(η) > 0 such that |g(t, x, z)| ≤ η|z(t, x)|, provided H0 (z(t)) < r(η).

Hence with the use of (14), (13) and (10) one gets that   1 d 1 V (t) = −ε −c zt2 dx dt 2 0  1  1   k β 2 + ck − cβ −ε zx dx + ε z 2 dx 2 2 0 0 − γ1 kzt (t, 0)2 − γ2 kzt (t, 1)2 εβ 2 z (t, 1) + εzt2 (t, 1). + εkγ22 zt2 (t, 1) − 2 Consequently, applying Lemma 1 one obtains that for any δ1 , δ2 > 0, and for all 0 < ε < kγ2 /(1 + kγ22 ) one has   1 1 d V (t) = −ε −c zt2 dx dt 2 0   1  k δ1 1 2 + ck − β + 2cβ − δ1 − δ2 −ε zx2 dx−ε z dx 2 2 0 0 β  δ2 − β + 2cβ − δ1 − δ2 z 2 (t, 1). − ε z 2 (t, 0) − ε 2 2 Choosing sufficiently small δ1 , δ2 > 0, and c ∈ (0, 0.5) sufficiently close to 0.5 one obtains that there exists C1 > 0 (independent of the initial conditions z0 and z1 ) such that   d V (t) ≤ −εC1 H0 (t) − εC1 z 2 (t, 0) + z 2 (t, 1) . dt Hence applying the last inequality in (15) one gets that for any sufficiently small ε > 0 there exists C2 ∈ (0, C1 ) (independent of z0 and z1 ) such that dV (t)/dt ≤ −εC2 V (t), which implies that V (t) ≤ V (0)e−εC2 t . Therefore, applying (15) again one gets that there exist M > 0 and δ > 0 (independent of z0 and z1 ) such that H0 (z(t)) ≤ M H0 (z(0))e−δt for any t ≥ 0. Here we used the fact that z 2 (0, 0) + z 2 (0, 1) ≤ KH0 (z(0)) for some K > 0 due to the fact that by the Sobolev imbedding theorem z(t)C[0,1] ≤ Cz(t)H 1 (0,1) for some C > 0. 

Let V (t) be as in Theorem 2. Then arguing in the same way as in the proof of this theorem one can check that for any sufficiently small ε > 0 there exist C2 > 0 and c ∈ (0, 0.5) such that  1 d V (t) ≤ −εC2 V (t) − zt g(t, x, z) dx dt 0  1  1 −ε xg(t, x, z)zx dx − εc zg(t, x, z) dx. (17) 0

0

Denote the function on the right-hand side of this inequality by W (t). As was pointed out above, for any η > 0 there exists r(η) > 0 such that |g(t, x, z)| ≤ η|z(t, x)|, and therefore   η 1 2 η 1 2 W (t) ≤ −εC2 V (t) + z dx + z dx 2 0 t 2 0  1  1  1 εη εη 2 2 + z dx + z dx + εcη z 2 dx 2 0 2 0 x 0   ε 1 + ε + 2εc ≤ −εC2 V (t) + η max 1, , H0 (z(t)). k β

if H0 (z(t)) < r(η). Consequently, applying the second inequality in (15) one obtains that there exists a sufficiently small η ∗ > 0 such that W (t) ≤ −

εC2 V (t), 2

(18)

provided H0 (z(t)) < r(η ∗ ). Suppose that H0 (z(t)) < r(η ∗ ) for all t ≥ 0. Then from (17) and (18) it follows that V (t) ≤ V (0)e−εC2 t/2 for all t ≥ 0, which with the use of (15) implies that there exist M > 0 and δ > 0 independent of z0 and z1 such that

Thus, the equilibrium point z(t, x) ≡ 0 of the linearized system (8)–(10) is globally exponentially stable in X = H 1 (0, 1) × L2 (0, 1). With the use of this result and Lyapunov’s indirect method we can (locally) extend this result to system (5)–(7). Theorem 3. Let f be continuously differentiable, f (0) = 0, and f  (0) > 0. Then for any γ1 ≥ 0 and γ2 > 0 there exist r > 0, M > 0 and δ > 0 such that for any z0 , z1 with z0 H1 (0,1) + z1 L2 (0,1) < r one has

H0 (z(t)) ≤ M H0 (z(0))e−δt ∗

∀t ≥ 0.

(19)

Choose r < r(η ) such that for any (z0 , z1 ) satisfying the inequality z0 H1 (0,1) + z1 L2 (0,1) < r one has max{1, M }H0 (z(0)) < r(η ∗ ). Let us verify that for any (z0 , z1 ) with z0 H1 (0,1) + z1 L2 (0,1) < r one has H0 (z(t)) < r(η ∗ ) for all t ≥ 0. Then (18) holds true for all t ≥ 0, and the proof is complete.

Arguing by reductio ad absurdum, suppose that there exists t ≥ 0 such that H0 (z(t)) ≥ r(η ∗ ). Define T = inf{t ≥ 0 : H0 (z(t)) ≥ r(η ∗ )}. Then, obviously, T > 0 and H0 (z(t)) < r(η ∗ ) for all t ∈ [0, T ). Hence with the use of (17) and (18) one obtains that (19) holds true for all t ∈ [0, T ). Therefore

H0 (z(t)) ≤ M H0 (z(0))e−δt ∀t ≥ 0, where z(t, x) is a solution of (5)–(7), i.e. the equilibrium point z(t, x) ≡ 0 of system (5)–(7) is exponentially stable in H 1 (0, 1) × L2 (0, 1).

Proof. Let z(t, x) be a solution of (5)–(7). For any t ≥ 0 and for a.e. x ∈ (0, 1) one has ztt (t, x) − kzxx (t, x) + βz(t, x) + g(t, x, z) = 0,

H0 (z(T )) ≤ M H0 (z(0))e−δT ≤ r(η ∗ )e−δT < r(η ∗ ),

which is impossible.  58

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57

4. ENERGY CONTROL PROBLEM

5. NUMERICAL STUDY

Let us also briefly discuss a more general control problem for system (1)–(4). Let Π(z) be a nonnegative continuously differentiable function such that Π(0) = 0. Define f (z) = Π (z), and consider the following initial-boundary value problem:   ztt (t, x) − kzxx (t, x) + f z(t, x) = 0, (20) z(0, x) = z0 (x), zt (0, x) = z1 (x), (21) (22) zx (t, 0) = u1 (t), zx (t, 1) = u2 (t).

System Description. In the numerical study of the energy problem for system (20)–(22), the function  control   f z(t, x) in (20) is taken as β sin z(t, x) , where β is some positive constant, so that (20)–(22) become the following sine-Gordon model with Neumann boundary conditions:   ztt (t, x) − kzxx (t, x) + β sin z(t, x) = 0, (27) z(0, x) = z0 (x), zt (0, x) = z1 (x), (28) (29) zx (t, 0) = u1 (t), zx (t, 1) = u2 (t). In this case, the Hamiltonian (23) turns to the form  1  2 zt zx2 H(t) = +k +β(1−cos z) dx (30) 2 2

As above, introduce the Hamiltonian   1 2 zx2 zt + k + Π(z) dx H(z) = 2 2 0

(23)

0

of equation (20). Clearly, H(z) is a nonnegative function, and it is preserved along solutions of the unforced system (i.e. when u1 (t) ≡ u2 (t) ≡ 0), since for any solution z(t, x) of (20)–(22) one has

In the present work, the function ψ(·) in (25), (26) is taken in the form of the saturation function, so that for numerical evaluation, control law (25), (26) has the form   u1 (t) = γ1 sat κ e(t) zt (t, 0), (31)   u2 (t) = −γ2 sat κ e(t) zt (t, 1), (32) where e(t) = H(t) − H ∗ denotes an error between the actual and reference values of H, κ > 0 is a control law parameter, sat(·) stands for the saturation function with limits {−1, 1}. For simplicity, controller gains γ1 , γ1 are taken equal to the common value γ, i.e. γ1 = γ2 = γ.

d H(z(t)) = kzt (t, 1)u2 (t) − kzt (t, 0)u1 (t). dt Furthermore, H(z(t)) = 0 for some t ≥ 0 iff Π(z(t, ·)) = 0 (which implies that f (z(t, ·)) = 0). In the case u1 (t) ≡ u2 (t) ≡ 0, one has H(z(t)) = 0 iff z(t, ·) = const is an equilibrium point of system (20)–(22) such that Π(z(t, ·)) = 0. Thus, H(z(t)) can be viewed as the system’s energy at time t.

The closed-loop energy control problem with plant model (27)–(29) and controller (30)–(32) was numerically studied by simulations in MATLAB/Simulink software environment. The following parameters have been used for the simulations: k = 0.12, β = 0.02. Initial conditions (28)   have been set to: z 0 (x) = A 1 − cos(2πx) , z 1 (x) = 0, where the magnitude parameter A was varied in different simulation runs. The gain κ = 100 is taken in (31), (32).

Choose a desired energy level H ∗ ≥ 0. We pose the following control problem: find a control law u(t) = (u1 (t), u2 (t)), which ensures the control objective H(z(t)) → H ∗ as t → +∞.

(24)

Thus, the control goal is to reach a desired energy level H ∗ in system (20)–(22). In the case, H ∗ = 0 this problem, in essence, is reduced to the stabilization problem.

Regulation Problem. In these simulation runs, the regulation problem is considered. To this end, desired energy level in (31), (32) is set to zero, H ∗ = 0, which gives the following form of the control law: u1 (t) = γ1 zt (t, 0), u2 (t) = −γ2 zt (t, 1). The initial magnitude parameter is taken as A = 5.

Let us design a control law with the use of the speedgradient algorithm (Fradkov et al., 1999; Fradkov, 2007).  2 Introduce the goal function Q(z(t)) = 12 H(z(t)) − H ∗ ) . Formally differentiating this function along solutions of (20)–(22) one obtains that    dQ(z(t)) = k H(z(t))−H ∗ ) zt (t, 1)u2 (t)−zt (t, 0)u1 (t) . dt Then according to the speed-gradient algorithm one can define  (25) u1 (t) = γ1 ψ H(z(t)) − H ∗ )zt (t, 0),  u2 (t) = −γ2 ψ H(z(t)) − H ∗ )zt (t, 1), (26)

For studying the dependence of the convergence rate of the energy of closed-loop control system on the controller gain γ, system equations (27)–(29), (30)–(32) have been numerically solved for various γ. The energy-control transient time t∗h was found as the minimal instant t such that {∀t ≥ t∗h : |e(t)| ≤ 0.05|e(0)|}. The result is depicted in Fig. 1. An interval of γ, where the energy control transient time is minimal was found.

where γ1 , γ2 ≥ 0 are scalar gains, and ψ : R → R is a function such that ψ(s) = 0 iff s = 0, and sψ(s) > 0 for any s = 0. Note that in the case H ∗ = 0 and ψ(s) = sign(s), control law (25), (26) is reduced to the one proposed in Section 2.

The evolution of system’s energy H(t) on time for γ = 3 is plotted in Fig. 2, showing asymptotic vanishing of system’s energy. The respective spatio-temporal plot of system state z(t, x) is depicted in Fig. 3. The results obtained are in correspondence with the theoretical statements.

In the case when the boundary condition on the left end has the form z(t, 0) = 0, the energy control problem posed above was analyzed in detail in (Dolgopolik et al., 2016, 2018). In the case of system (20)–(22) this problem, as well as the analysis of control law (25), (26), remain challenging open problems. Below, we only present some promising results of numerical experiments.

To demonstrate an advantage of two-point boundary control in comparison with the single-point one, stated in authors’ previous work (Dolgopolik et al., 2018), the simulations were made for both cases γ1 = γ2 = 3 (twopoint control) and γ1 = 0, γ3 = 3 (single-point control). The result is depicted in Fig. 2. It is seen from the plot 59

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Energy Pumping. The simulation for the case when the desired energy H ∗ is greater than the initial one H(0), has been made to illustrate the control system (27)–(29), (30)– (32) performance for an energy pumping (“heating”) case. For simulation, A = 1.0, H ∗ = 10, γ = 1 are taken. The simulation results are plotted in Figs. 4, 5, demonstrating achievement of the control goal. (It can be shown that for the case of energy pumping, the control goal may be achieved in the finite-time if the function sat(·) in (31), (32) is substituted by the discontinuous function sign(·).)

that the two-point control (solid line on the plot) ensures significantly faster achievement of the control goal than the single-point one (dashed line). t* 8

h

7 6 5

H(t), H*

4 10

3

H H*

2 8 1 6

0 1

2

3

4

5

6

7

8 9 10 4

Fig. 1. Dependence of regulation transient time t∗h on γ. 2

H(t) 30

0 0

25

0.5

1

1.5

2

2.5

3

3.5

4

t

Fig. 4. Time history of system’s energy H(t) for A = 1.0, H ∗ = 10, γ = 1.

20 15 10 5 0 0

0.5

1

1.5

2

2.5

3

3.5

4

t

Fig. 2. Time histories of system’s energy H(t) for twopoint boundary control, γ1 = γ2 = 3 (solid line), and single-point boundary control, γ1 = 0, γ2 = 3 (dashed line).

Fig. 5. Spatio-temporal plot of system’s state z(t, x) for A = 1.0, H ∗ = 10, γ = 1. 6. CONCLUSION In this paper a new control problem is posed and solved: the regulation problem for the one-dimensional KleinGordon and semilinear wave equation with Neumann boundary conditions in the case when the control acts at both ends of the space interval (“two-point control”). The proposed control algorithm is based on the speed-gradient method. The global exponential stability of the closed loop system for the case of the Klein-Gordon equation is established by means of a new Lyapunov functional. This result is extended to the case of the semilinear wave equation by means of linearization.

Fig. 3. Spatio-temporal plot of system’s state z(t, x) for γ = 3. 60

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In addition, the two-point energy control problem for the sine-Gordon and semilinear wave equation was addressed by means of simulation. It is demonstrated that the proposed two-point control algorithm may provide 30% faster transients.

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