By Panos J. Antsaklis

In keeping with a streamlined presentation of the authors’ winning paintings Linear structures, this textbook offers an creation to structures thought with an emphasis on regulate. preliminary chapters current worthwhile mathematical historical past fabric for a basic realizing of the dynamical habit of structures. each one bankruptcy comprises worthy bankruptcy descriptions and directions for the reader, in addition to summaries, notes, references, and routines on the finish. The emphasis all through is on time-invariant platforms, either non-stop- and discrete-time.

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94). Conservative dynamical systems are described by the system of 2n ordinary diﬀerential equations ∂H (p, q), i = 1, . . , n, ∂pi ∂H p˙ i = − (p, q), i = 1, . . , n. 96) [along the solutions qi (t), pi (t), i = 1, . . , n], then we obtain, by the chain rule, Exercises dH (p(t), q(t)) = dt n i=1 n ∂H (p, q)p˙i + ∂pi n i=1 ∂H (p, q)q˙i ∂qi ∂H ∂H = − (p, q) (p, q) + ∂pi ∂qi i=1 n =− i=1 35 n i=1 n ∂H ∂H (p, q) (p, q) ∂qi ∂pi ∂H ∂H ∂H ∂H (p, q) (p, q) + (p, q) (p, q) ≡ 0. 96). This constant is determined by the initial data (p(0), q(0)).

Let f ∈ C(D, Rn ). 12) has a solution deﬁned on [t0 , t0 + c) for some c > 0. 6]. 12). 9. Let f ∈ C(D, Rn ). 42) for all (t, x), (t, y) ∈ K where LK > 0 is a constant depending only on K. 12) has at most one solution on any interval [t0 , t0 + c), c > 0. 8]. 42) is automatically satisﬁed. 11) on an interval J. 11) on J0 . Then φ is said to be continued or extended to the larger interval J0 . When no such continuation is possible, then φ is called noncontinuable. 10. 43) 1 deﬁned on J = (−1, 1).

Two link pendulum resistors in electric circuits, then we can use Lagrange’s equation to describe such systems. ) For a system with n degrees of freedom, this equation is given by ∂D ∂L d ∂L (q, q) ˙ + (q, q) ˙ − (q) ˙ = fi , i = 1, . . 97) dt ∂ q˙i ∂q ∂ q˙i where q T = (q1 , . . , qn ) denotes the generalized position vector. , the diﬀerence between the kinetic energy T and the potential energy W . , [βij ] is symmetric and all of its eigenvalues are nonnegative). The dissipation function D represents onehalf the rate at which energy is dissipated as heat.