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Differential Calculus of Vector Functions of one Variable
Published in C. Young Eutiquio, Vector and Tensor Analysis, 2017
Let us first recall the definition of a real-valued (scalar) function in a domain D. We note that a domain D is an open set of points in which any two points can be joined by line segments lying entirely in the set. Examples are the set of points inside a sphere or a parallelepiped in three dimensional space, the set of points inside a circle or a rectangle in two dimensional space, and the set of points in the interval a < x < b in one dimension. Now we recall that a real-valued function defined in a domain D is a mapping or a rule f that assigns to each point P in D a unique real number f(P) from a set R. We call D the domain of definition of f and R the range of f. In a similar manner, we define a vector-valued function or simply a vector function in a domain D as a rule F that assigns to each point P in D a unique vector denoted by F(P). In other words, a function defined in a domain D whose range is a set of vectors is called a vector function. In this chapter, we are concerned with vector functions defined in an interval which may contain one or both endpoints or which may be infinite. Such functions are called vector functions of a real variable. The real variable is usually denoted by t, which indicates time in many applications. We denote vector functions by bold-faced letters F, G, etc., and the value of a function F at t by F(t).
Vector and Tensor Calculus
Published in Tasos C. Papanastasiou, Georgios C. Georgiou, Andreas N. Alexandrou, ViscousFluid Flow, 2021
Tasos C. Papanastasiou, Georgios C. Georgiou, Andreas N. Alexandrou
Vectors are specified by their magnitude and their direction with respect to a given frame of reference. They are often denoted by lower case, boldface type, such as u for the velocity vector. A vector field is a vector-valued function that associates a vector with each point of a given region in space. For example, the velocity of the fluid in the region V of Fig. 1.1 defines a vector field denoted by u(x, y, z, t). A vector field which is independent of time is called a steady-state or stationary vector field. The magnitude of a vector u is designated by lul or simply by u.
Distributed Parameter Nonlinear Transmission Lines II: Existence of Weak Solutions
Published in Frederick Bloom, Mathematical problems of classical nonlinear electromagnetic theory, 2020
D(f) denoting, of course, the domain of the vector-valued function f. If we now take the inner product, in Rm, of both sides of (5.4.3) with ∇η (uε) we obtain the equation η(uε)t+q(uε)x−εη(uε)xx+ε∇2η(uε)[uxε,uxε]=0 where ∇2η denotes the Hessian matrix associated with the entropy function η(·), while ∇2η(uε)[uxε,uxε]=(uxε)t⋅(uε)⋅uxε the superscript “t” in (5.4.8) denoting the transpose of the indicated vector. If, for the solution of (5.4.7), we could show that uεk → ū (strong convergence, a.e.), as k → ∞, it would then follow that ū would satisfy the entropy inequality η(u¯)t+q(u¯)x≤0
Robust output consensus of networked negative imaginary systems via an integral quadratic constraint approach
Published in International Journal of Control, 2023
Qian Zhang, Liu Liu, Yufeng Lu
In this section, the homogeneous and heterogeneous networks of n NI systems with disturbances are shown and the robust output consensus problem is given. denotes the square integrable vector-valued function space with a corresponding dimension. denotes the -valued Hardy space in . The paper Skeik et al. (2019) gives some results under the directed topology, however, the NI system is very special. This paper considers generalising NI systems in Mabrok et al. (2014a). The directed network topology can not satisfy some requirements which we need. In addition, there is no unified framework to solve the robust output consensus of networked NI systems with the NI definition in Mabrok et al. (2014a) under switching interaction topologies. For instance, there is a big difference between consensus problems with single-integrator and double-integrator dynamics under switching interaction topologies, more details see Ren and Beard (2008). In order to get framing results we do not consider switching interaction topologies. Hence, we assume that the information flow among NI systems is modelled by an undirected and connected graph in this paper.
Parameter estimation for models of chemical reaction networks from experimental data of reaction rates
Published in International Journal of Control, 2023
Manvel Gasparyan, Arnout Van Messem, Shodhan Rao
We commence by introducing the notations that are used throughout the manuscript. For any vector , denotes its ith element, i.e. . Denote by the diagonal matrix, whose diagonal entries are the elements of the vector v. denotes the entry of the matrix M corresponding to the ith row and the jth column. Denote by the identity matrix. Define the vector-valued function as , and the vector-valued function as . For a vector-valued differentiable function given as , denote the partial Jacobian matrix with respect to its first variable x with . Similarly, denote the partial Jacobian matrix of f with respect to its second variable y with .
Stability and L 1 × ℓ1-to-L 1 × ℓ1 performance analysis of uncertain impulsive linear positive systems with applications to the interval observation of impulsive and switched systems with constant delays
Published in International Journal of Control, 2020
Let us assume here that the system (1) is internally positive and that the matrices of the system are such that they remain constant for all values . Assume further that there exist a differentiable vector-valued function a vector-valued function a vector and scalars such that the conditions and hold for all . Then, the system (1) is asymptotically stable for all and under the minimum dwell-time condition . Moreover, the mapping has a hybrid -gain of at most γ.