TY - GEN

T1 - How to establish the exact controllability of nonlinear DPS using iterations back and forth in time

AU - Natarajan, Vivek

AU - Zhou, Hua Cheng

AU - Weiss, George

N1 - Publisher Copyright:
© 2017 American Automatic Control Council (AACC).

PY - 2017/6/29

Y1 - 2017/6/29

N2 - Consider a nonlinear distributed parameter system (DPS) described by (t) = Ax(t)+Bu(t)+BNN(x(t), t) for all t ≥ 0. Here A is the infinitesimal generator of a strongly continuous group of operators T on a Hilbert space X, B and BN, defined on Hilbert spaces U and UN, respectively, are admissible control operators for T and N : X × [0,∞) → UN is continuous in t and Lipschitz in x, with Lipschitz constant LN independent of t. Thus B and BN can be unbounded as operators from U and UN to X, in which case the nonlinear term BNN(x(t), t) in the DPS is in general not a Lipschitz map from X ×[0,∞) to the state space X. Our goal is to find conditions under which this DPS is exactly controllable in some time τ, which means that for any initial state x(0) ϵ X, we can steer the final state x(τ) of the DPS to any chosen point in X by using an appropriate input function u ϵ L2([0, τ];U). We suppose that there exist linear operators F and Fb such that (A, [B BN], F) and (-A; [B BN], Fb) are regular triples and A + BFΛ and -A + BFb,Λ are generators of strongly continuous semigroups Tf and Tb on X such that ∥Ttf∥ · ∥Ttb∥ decays to zero exponentially. We prove that if LN is sufficiently small, then the nonlinear DPS is exactly controllable in some time τ > 0. Our proof is constructive and provides a numerical algorithm for approximating the required control signal. We illustrate our approach using a simple example.

AB - Consider a nonlinear distributed parameter system (DPS) described by (t) = Ax(t)+Bu(t)+BNN(x(t), t) for all t ≥ 0. Here A is the infinitesimal generator of a strongly continuous group of operators T on a Hilbert space X, B and BN, defined on Hilbert spaces U and UN, respectively, are admissible control operators for T and N : X × [0,∞) → UN is continuous in t and Lipschitz in x, with Lipschitz constant LN independent of t. Thus B and BN can be unbounded as operators from U and UN to X, in which case the nonlinear term BNN(x(t), t) in the DPS is in general not a Lipschitz map from X ×[0,∞) to the state space X. Our goal is to find conditions under which this DPS is exactly controllable in some time τ, which means that for any initial state x(0) ϵ X, we can steer the final state x(τ) of the DPS to any chosen point in X by using an appropriate input function u ϵ L2([0, τ];U). We suppose that there exist linear operators F and Fb such that (A, [B BN], F) and (-A; [B BN], Fb) are regular triples and A + BFΛ and -A + BFb,Λ are generators of strongly continuous semigroups Tf and Tb on X such that ∥Ttf∥ · ∥Ttb∥ decays to zero exponentially. We prove that if LN is sufficiently small, then the nonlinear DPS is exactly controllable in some time τ > 0. Our proof is constructive and provides a numerical algorithm for approximating the required control signal. We illustrate our approach using a simple example.

UR - http://www.scopus.com/inward/record.url?scp=85027011527&partnerID=8YFLogxK

U2 - 10.23919/ACC.2017.7963326

DO - 10.23919/ACC.2017.7963326

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AN - SCOPUS:85027011527

T3 - Proceedings of the American Control Conference

SP - 2485

EP - 2490

BT - 2017 American Control Conference, ACC 2017

PB - Institute of Electrical and Electronics Engineers Inc.

T2 - 2017 American Control Conference, ACC 2017

Y2 - 24 May 2017 through 26 May 2017

ER -