## Logic-Based Knowledge Representation by Baader F

By Baader F

A bankruptcy from

M.J. Wooldridge and M. Veloso (Eds.) - synthetic Intelligence this day, Springer-Verlag, 1999 (LNAI 1600) (pp. 13-41)

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23) 0 λ (t) ⊥ w(t) = −x2 (t) 0 . 15), it is not possible to calculate λ (t) directly from this set of relations. At first sight there is no LCP that can be constructed (indeed now we have a zero matrix M). Let us, however, imagine that there is a time interval [τ , τ + ε ), ε > 0, on which the solution x2 (t) = 0 for all t ∈ [τ , τ + ε ). Then on [τ , τ + ε ) one has necessarily −x˙2 (t) 0, otherwise the unilateral constraint −x2 (t) 0 would be violated. Actually all the derivatives of x2 (·) are identically 0 on [τ , τ + ε ).

1 Explicit Time-Stepping Schemes Based on ODE with Discontinuities Formulations A forward Euler scheme may be applied on an ODE with discontinuities of the form, x˙ = f (x,t), where the right-hand side may possess discontinuities (see Sect. 8). 76c) for −x2 > 0. ⎪ ⎩ − x2 − 1 x1 − Vz L LC The simulation for this choice of the right-hand side is illustrated in Fig. 12. 76b) cannot be reached due to the numerical approximation on x2 . This artifact results in spurious oscillations of the diode voltage v(t) = λ (t) and the diode current x2 (t) = ω (t) as we can observe on the zoom in Fig.

24) which is an LCP to be solved only when x2 (t) = 0. The fact that this LCP possesses a solution λ (t) − x˙2(t) > 0 is a sufficient condition for the system to change its mode of evolution. 15). 1 Electrical Circuits with Ideal Diodes 13 LCPs with a unique solution. 24). 7). We obtain that λ (t) ∈ −∂ ψIR+ (−x2 (t)). 25) where it is implicitly assumed that x2 (0) 0 so that the inequality constraint x2 (t) 0 will be satisfied for all t 0. 25) can be viewed similarly as the index-reduction operation in a Differential Algebraic Equation (DAE).