Control and Nonlinearity
نویسندگان
چکیده
linear control system, 51-67Cauchy problem, 52-54controllability, 55-62Korteweg-de Vries equation, 65—67transport equation, 62-65adjoint operator, 373a d ^ y (definition), 130admissibility condition, 52, see also regularity propertyapproximate controllability, 55, 57, 61Ascoli's theorem, 152, 170attractor, 280averaging, 332-333 backstepping, 334-337, 353desingularization, 335-336, 353Banach fixed-point theorem, 163Brockett condition (obstruction tostabilizability), 289, 292, 297, 341, 357,361Brouwer fixed-point theorem, 133, 236, 244,390, 391Burgers equation, 202, 360 Carleman inequality, 80, 113Cayley-Hamilton theorem, 10, 133characteristic curves, 363characteristic polynomial, 10, 275characteristic velocities, 370closed operator, 373closed-loop system (definition), 281compact resolvent, 95, 228, 354companion matrix, 276 compatibility conditions, 165, 200, 365, 366control affine (definition), 131control Lyapunov function, 313—315, 334,336, 353, 355application to stabilization, 313-314controllability abstract linear control system, 55-62Euler equations, 194-197finite-dimensional linear control system,3-22finite-dimensional nonlinear controlsystem, 125-157heat equation, 103-118, 225-233hyperbolic system, 165-174Korteweg-de Vries equation, 42-49Navier-Stokes equations, 197-203Saint-Venant equations, 204-221Schrodinger equation, 96-99, 248-270spacecraft, 128-129, 144-145theoremlinear control system, 55-62Euler equations, 194-197finite-dimensional linear control system,3-22finite-dimensional nonlinear controlsystem, 125-157heat equation, 103-118, 225-233hyperbolic system, 165-174Korteweg-de Vries equation, 42-49Navier-Stokes equations, 197-203Saint-Venant equations, 204-221Schrodinger equation, 96-99, 248-270spacecraft, 128-129, 144-145theorem abstract linear control system, 56, 57Euler equations, 195-197heat equation, 79, 87, 104, 105, 225hyperbolic system, 168Kalman rank condition, 9Korteweg-de Vries equation, 42, 161,237linear finite-dimensional controlsystem, 6, 9, 11Navier-Stokes equations, 199, 201necessary condition, 134, 145Saint-Venant equations, 204Schrodinger equation, 96, 248, 251sufficient condition, 135, 143transport equation, 29transport equation, 29-37controllability Gramian, 6, 151, 282controller form, 275Couette flows, 232CRONE, 281linear control system, 56, 57Euler equations, 195-197heat equation, 79, 87, 104, 105, 225hyperbolic system, 168Kalman rank condition, 9Korteweg-de Vries equation, 42, 161,237linear finite-dimensional controlsystem, 6, 9, 11Navier-Stokes equations, 199, 201necessary condition, 134, 145Saint-Venant equations, 204Schrodinger equation, 96, 248, 251sufficient condition, 135, 143transport equation, 29transport equation, 29-37controllability Gramian, 6, 151, 282controller form, 275Couette flows, 232CRONE, 281 damping, 314-328definitiona d ^ y , 130adjoint operator, 373approximate controllability, 55closed operator, 373closed-loop system, 281control affine, 131control Lyapunov function, 313controllability Gramian, 6controllability of x = A{t)x + B(t)u, 4degree of a function, 379dissipative operator, 373
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