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Published in S.P. Bhattacharyya, L.H. Keel, of UNCERTAIN DYNAMIC SYSTEMS, 2020
Edmond A. Jonckheere, Jonathan R. Bar-on
Theorem 1.Assume m(.) is twice continuously differentiable and that D × Ω is a differentiable manifold. Thenm−1(∂2N)⊆{Morsecriticalpointsoffθ:θ∈[0,2π]}
Computer-Aided Analysis of Nonlinear Systems
Published in Derek A. Linkens, CAD for Control Systems, 2020
Under natural conditions on f, the set of solutions of Eq. (5) constitutes a differentiable manifold in the product of state space and parameter space. The dimension of this manifold equals that of the parameter space. At present, the standard path following computational methods require the user to construct a picture of a p-dimensional manifold from information along a one-dimensional path [50].
Tools for Ordinary Differential Equations Analysis
Published in Wilfrid Perruquetti, Jean-Pierre Barbot, Chaos in Automatic Control, 2018
In this expression: t∈ℐ⊂ℝ represents the time variable and X is the state space.1 For practical reasons, the state space may be bounded in order to take into account physical limitations. In general, the state space is a differentiable manifold. When the vector x contains a variable and its successive derivatives, X is then called phase space. However, some authors (p. 11 of [2]) use the two designations without discrimination. The vector x∈X is the state vector of (2.10) (sometimes the phase vector according to the situation). In practice, it contains a sufficient number of variables useful to describe the time evolution of the process; x(t) is the instantaneous state at time t and f:ℐ×X→TX (tangent space), (t, x) ↦ f (t, x), is the vector field. To simplify the rest of the presentation, we will consider the particular case where ℐ×X is an open of ℝn+1 and TX is ℝn.
Tseng's extragradient algorithm for pseudomonotone variational inequalities on Hadamard manifolds
Published in Applicable Analysis, 2022
Jingjing Fan, Xiaolong Qin, Bing Tan
Let be a finite dimensional differentiable manifold. The set of all tangents at is called a tangent space of at , which forms a vector space of the same dimension as and is denoted by . The tangent bundle of is denoted by , which is naturally a manifold. We denote by the scalar product on with the associated norm , where the subscript x is sometimes omitted. A differentiable manifold with a Riemannian metric is called a Riemannian manifold. Let be a piecewise differentiable curve joining to in , we can define the length of . The minimal length of all such curves joining x to y is called the Riemannian distance and it is denoted by .
Karush-Kuhn-Tucker optimality conditions for non-smooth geodesic quasi-convex optimization on Riemannian manifolds
Published in Optimization, 2023
Feeroz Babu, Akram Ali, Ali H. Alkhaldi
Let be a differentiable manifold of dimension n and be a tangent space of at . Let be the tangent bundle of . For every , tangent space forms a vector space of the same dimension as . We consider the n-dimensional differentiable real manifold , i.e. for every . A smooth mapping is said to be a Riemannian metric on , if for every , is an inner product on . The induced norm is given by for every and every . For simplicity and convenience, we shall omit the subscript x. A differentiable manifold endowed with a Riemannian metric is said to be a Riemannian manifold. Let be a piecewise smooth curve such that and , then Riemannian distance is given by where denotes the tangent vector at . The open metric ball in centred at with radius denoted by , is defined as If is parallel along γ, i.e. for all , then γ is said to be a geodesic and in this case is a constant. Moreover, if , then γ is called normalized geodesic. We denote the set of all minimal geodesics with and by , is defined as where is the zero vector in tangent space and ▽ is a Levi-Civita connection.