Lipschitz Continuous Ordinary Differential Equations are Polynomial-Space Complete
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In answer to Ko’s question raised in 1983, we show that an initial value problem given by a polynomial-time computable, Lipschitz continuous function can have a polynomial-space complete solution. The key insight is simple: the Lipschitz condition means that the feedback in the differential equation is weak. We define a class of polynomial-space computation tableaux with equally weak feedback, and show that they are still polynomial-space complete. The same technique also settles Ko’s two later questions on Volterra integral equations.
Keywords.Computable analysis computational complexity initial value problem Lipschitz condition ordinary differential equations Picard–Lindelöf theorem polynomial space
Subject classification.03F60 68Q17 65Y20 65L05 03D15
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