{"product_id":"9781470451387-tunneling-estimates-and-approximate","title":"Tunneling Estimates and Approximate Controllability for Hypoelliptic Equations","description":"\u003cmeta content=\"text\/html; charset=utf-8\" http-equiv=\"Content-Type\"\u003e\u003cp\u003e\u003cspan\u003eThis memoir is concerned with quantitative unique continuation estimates for equations involving a ‘sum of squares’ operator L on a compact manifold M assuming: (i) the Chow-Rashevski-Hormander condition ensuring the hypoellipticity of L,and(ii) the analyticity of M and the coefficients of L.\u003cbr\u003eThis memoir is concerned with quantitative unique continuation estimates for equations involving a \"sum of squares\" operator L on a compact manifold M assuming: (i) the Chow-Rashevski-H¨ormander condition ensuring the hypoellipticity of L,and(ii) the analyticity of M and the coefficients of L. The first result is the tunneling estimate ?L2(?) ? Ce?c?k 2 for normalized eigenfunctions ? of L from a nonempty open set ? ?M,wherek is the hypoellipticity index of L and ? the eigenvalue. The main result is a stability estimate for solutions to the hypoelliptic wave equation (?2 t + L)u =0:forT\u0026gt;2supx?M(dist(x,?)) (here, dist is the subRiemannian distance), the observation of the solution on (0,T) × ? determines the data. The constant involved in the estimate is Cec?k where?isthetypical frequency of the data. Wethen prove the approximate controllability of the hypoelliptic heat equation (?t +L)v = 1?f in any time, with appropriate (exponential) cost, depending on k. In case k = 2 (Grushin, Heisenberg.), we further show approximate controllability to trajectories with polynomial cost in large time. We also explain how the a nalyticity assumption can be relaxed, and a boundary ?Mcan be added in some situations.\u003cbr\u003e\u003cbr\u003e\u003c\/span\u003e\u003c\/p\u003e","brand":"Rarewaves","offers":[{"title":"Default Title","offer_id":54845446717814,"sku":"9781470451387","price":68.38,"currency_code":"GBP","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0092\/7504\/8033\/files\/orig_38185655_57380dbf-414f-4160-9c86-d3c348b2180b.jpg?v=1763787301","url":"https:\/\/www.rarewaves.com\/products\/9781470451387-tunneling-estimates-and-approximate","provider":"Rarewaves.com","version":"1.0","type":"link"}