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In the limit A → ∞ (representing an ideal operational amplifier), the third model equation must be replaced by x2 = 0. In this case, the above system can be reduced to ˙ x3 = −RC u(t). Of course, for this equation there cannot exist a function as limit for increasing γ . Thus, in order to treat such problems we need solution spaces that are more general than spaces of functions. , [199]. 3. Behavior of a non-ideal differentiator — input profiles C and compact support in R. We say that a sequence (φi (t))i∈N converges to zero in D and write φi → 0, if all functions vanish outside the same bounded interval (q) and the sequences (φi )i∈N of the q-th derivatives converge uniformly to zero for all q ∈ N0 .

Consider the system in distributional form 1 0 0 0 1 0 x˙1 = 0 1 x˙2 x1 f 1 0 + 1 + 0 0 x2 f2 x1,0 δ. x2,0 with f− = 0. Note that this differential-algebraic equation has index ν = 1. Requiring that x− = 0, we obtain x2 = −f2 and x1 solves x˙ 1 = x1 + f1 + x1,0 δ. If iord f ≤ −1, then both components can be associated with functions, regardless of the choice of the initial condition. 50 says that all vectors x0 are weakly consistent in this case. 74) reads x1 − x4 = H, C(x˙1 − x˙2 ) + x3 − x2 = C(x1,0 − x2,0 )δ, R x2 = 0, x4 = 0.

RC RC Let the input voltage u be defined by u(t) = u0 > 0 for t < 0 and u(t) = 0 for t ≥ 0. 1. Discharging a capacitor between x1 and x3 . As initial condition, we take x2 (0) = u0 . The differential equation can then be solved separately for t < 0 and t > 0. Since both parts can be joined together to a continuous function, we may view x2 defined by x2 (t) = u0 u0 e−t/RC for t < 0, for t ≥ 0. 34 2 Linear differential-algebraic equations with constant coefficients as solution everywhere in R. This procedure can be formalized for linear differential equations working with piecewise continuous inhomogeneities and continuous, piecewise continuously differentiable solutions.

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