Esistenza,unicità e calcolo della soluzione di un sistema non lineare

LetX be a point of the realn-dimensional Euclidean space ? ⁿ ,G(X) a given vector withn real components defined in ? ^u ,U an unknown vector withs real components,K a known vector withs real components andA a given reals×n matrix of ranks. Assuming that, for every pair of pointsX ₁ , X₂of ? ⁿ ,G(X) satisfies the conditions $$(G(X_1 ) - G(X_2 ), X_1 - X_2 ) \geqslant o (X_1 - X_2 , X_1 - X_2 )$$ and $$\left\| {(G(X_1 ) - G(X_2 )\left\| { \leqslant M} \right\|X_1 - X_2 )} \right\|$$ wherec andM are positive constants, we prove that a unique solution of the system $$\left\{ \begin{gathered} G(X) + A ^T U = 0 \hfill \\ AX = K \hfill \\ \end{gathered} \right.$$ exists and we show a method for finding such a solution     

On the transversality of functions at the core of the transverse function approach to control

The transverse function approach to control, introduced by Morin and Samson in the early 2000s, is based on functions that are transverse to a set of vector fields in a sense formally similar to, although strictly speaking different from, the classical notion of transversality in differential topology. In this paper, a precise link is established between transversality and the functions used in the transverse function approach. It is first shown that a smooth function ${f : M \longrightarrow Q}$ is transverse to a set of vector fields which locally span a distribution D on Q if, and only if, its tangent mapping T f is transverse to D, where D is regarded as a submanifold of the tangent bundle T Q. It is further shown that each of these two conditions is equivalent to transversality of T f to D along the zero section of T M. These results are then used to rigorously state and prove that if M is compact and D is a distribution on Q, then the set of mappings of M into Q that are transverse to D is open in the strong (or ??Whitney C ^??-??) topology on C ^??(M, Q).     

Local controllability of non-linear systems: An example

An example of a locally controllable nonlinear system on R³ is given. The system is of the type with f, g analytic vector fields and u bounded. The fields f and g are such that the dimension of the vector space spanned at x₀ by the Lie brackets which contain g an odd number of times is 2.     

Controllability for single-input mechanical control systems with dissipation

Within the affine connection framework of Lagrangian control systems , based on the results of Sussmann on small- time locally controllability of single- input affine nonlinear control systems , the controllability results for mechanical control systems with single- input are extended to the case of the systems with isotropic damping ,where the Lagrangian is the kinetic energy associated with a Riemannian metric. A sufficient condition of negative small- time locally controllability for the system is obtained. Then ,it is demonstrated that such systems are small- time locally configuration controllable if and only if the dimension of the configuration manifold is one. Finally , two examples are given to illustrate the results. Lie bracketting of vector fields and the symmetric product show the advantages in the discussion.     

Controllability for single-input mechanical control systems with dissipation

Within the affane connection framework of Lagrangian control systems, based on the results of Sussmann on small-time locally controllability of single-input affine nonlinear control systems, the controllability results for mechanical control systems with single-input are extended to the case of the systems with isotropic damping, where the Lagrangian is the kinetic energy associated with a Riemannian metric, A sufficient condition of negative small-time locally controllability for the system is obtained.Then,it is demonstrated that such systems are small-time locally configuration controllable if and only if the dimemion of the configuration manifold is one. Finally, two examples are given to illustrate the results. Lie bracketting of vector fields and the symmetric product show the advantages in the discussion.     

Controllability of general nonlinear systems

Consider the nonlinear system $$\dot x(t) = f(x(t)) + \sum\limits_{i = 1}^m {u_i (t)g_i (x(t)), x(0) = x_0 \in M}$$ whereM is aC ^∞ realn-dimensional manifold,f, g ₁,?.,g _m areC ^∞ vector fields onM, andu ₁ ,..,u _m are real-valued controls. Ifm=n?1 andf, g ₁ ,?,g _m are linearly independent, then the system is called a hypersurface system, and necessary and sufficient conditions for controllability are known. For a generalm, 1 ≤m ≤n?1, and arbitraryC ^∞ vector fields,f, g ₁ ,?,g _m , assume that the Lie algebra generated byf, g ₁ ,?,g _m and by taking successive Lie brackets of these vector fields is a vector bundle with constant fiber (vector space) dimensionp onM. By Chow's Theorem there exists a maximalC ^∞ realp-dimensional submanifoldS ofM containingx ₀ with the generated bundle as its tangent bundle. It is known that the reachable set fromx ₀ must contain an open set inS. The largest open subsetU ofS which is reachable fromx ₀ is called the region of reachability fromx ₀. IfO is an open subset ofS which is reachable fromx ₀,S we find necessary conditions and sufficient conditions on the boundary ofO inS so thatO = U. Best results are obtained when it is assumed that the Lie algebra generated byg ₁,?,g _m and their Lie brackets is a vector bundle onM.     

The convex-decomposable operator equation and its monotonic inclusive iteration

In this paper we discuss a class of nonlinear operators, that we name convex-decomposable operators, which have the nature:Fy?Fx≤(D ₁ (x, y)+D ₂ (x, y))(y?x) forx≤y, andD ₁ andD ₂ are isotone and antitone respectively. It is shown that it is a rather large class of operators, containing e.g. the operators whose second derivatives are bounded. For these operators we give a monotonic inclusive iterative algorithm, and show the convergence of this algorithm under some simple conditions.     

Distributions having bases which generate finite dimensional Lie algebras

Let X¹,…, X^k be real analytic vector fields on an n-dimensional manifold M, k < n, which are linearly independent at a point p ε M and which, together with their Lie products at p, span the tangent space TM_p. Then X¹,…, X^k form a local basis for a real analytic k-dimensional distribution x→D^k(x)=span{X¹(x),…,X^k(x)}. We study the question of when D^k admits a basis which generates a nilpotent, or solvable (or finite dimensional) Lie algebra. If this is the case the study of affine control systems, or partial differential operators, described via X¹,…, X^k can often be greatly simplified.     

Global controllability of nonlinear systems in two dimensions

LetM be a connected real-analytic 2-dimensional manifold. Consider the system (t) = f(x(t)) + u(t)g(x(t)),x(0) =x ₀ M, wheref andg are real-analytic vector fields onM which are linearly independent at some point ofM, andu is a real-valued control. Sufficient conditions on the vector fieldsf andg are given so that the system is controllable fromx ₀. Suppose that every nontrivial integral curve ofg has a pointp wheref andg are linearly dependent,g(p) is nonzero, and that the Lie bracket [f,g] andg are linearly independent atp. Then the system is controllable (with the possible exception of a closed, nowhere dense set which is not reachable) from any pointx ₀ such that the vector space dimension of the Lie algebraL _A generated byf,g and successive Lie brackets is 2 atx ₀.Research supported in part by the National Science Foundation under NSF Grant MCS76-05267-A01 and by the Joint Services Electronics Program under ONR Contract 76-C-1136.     

Local controllability of scalar input systems on 3-manifolds

Local controllability of generic C^k pairs of vector fields in a three dimensional connected manifold M is studied, showing that in general a necessary condition is also sufficient on an open dense subset of the set of points where it is verified. The results obtained are used to determine local controllability for scalar input affine control systems.     

Spurious solutions for discrete superlinear boundary value problems

We consider finite dimensional nonlinear eigenvalue problems of the typeAu=λFu whereA is a matrix and(Fu) _i =f(u _i ),i=1, ...,m. These may be thought of as discretizations of a corresponding boundary value problem. We show that positive, spurious solution branches of the discrete equations (which have been observed in some cases in [1, 7]) typically arise iff increases sufficiently strong and ifA ^?1 has at least two positive columns of a certain type. We treat in more detail the casesf(u)=e ^u andf(u)=u ^α where also discrete bifurcation diagrams are given.     

Improved approximations for the minimum-cut ratio and the flux

LetC be any concurrent flow problem with integer demands and capacities on an undirected graphG. Kleinet al. [5] recently showed that the maximum throughputz ^* is Ω (S/(logC logD)), whereS is the minimum cut ratio,C is the sum of the capacities on the edges, andD is the sum of the demands of the commodities. They also presented a polynomial time algorithm which finds a cut whose ratio isS · O(logC logD). Leighton and Rao [8] have shown that for the concurrent flow problem with a unit demand between every pair of nodes,z ^* is Ω (S/logn), wheren is the number of nodes ofG. This leads to anO(logn) approximation of the flux of a graphG with uniform weights on the nodes. We show that for the problem in [5] the maximum throughputz ^* is Ω(S/(logn logD)), and we find a cut whose ratio isS · O (logn logD). For the special case in which the demands have the form $$demand between u and v = l(u) \cdot \frac{{l(v)}}{2},$$ wherel(u) is an assignment of a positive integer weight to nodeu, we show that the lower bound onz* can be improved to be Ω(S/logn). This generalizes the result of Leighton and Rao.     

Bang-bang strategies using interrupted observations for steering a random motion of a point

Consider the random motion in the plane of a point M, whose velocity u = (u₁, u₂) is perturbed by an R²-valued Gaussian white noise. The point is equipped with an observation mechanism which provides interrupted noisy observations of a given target. The interruptions are characterized by a jump Markov process taking on the values 0 or 1, and using these observations the point M wishes to steer itself into the target. Sufficient conditions on optimal velocity laws (strategies) u are derived, and suboptimal strategies computed. The numerical results obtained show that the suboptimal strategies obey the separation principle of optimal stochastic control theory.     

The existence of positive solutions of singular fractional boundary value problems

We discuss the existence of positive solutions for the singular fractional boundary value problem D^αu+f(t,u,u^′,D^μu)=0, u(0)=0, u^′(0)=u^′(1)=0, where 2<α<3, 0<μ<1. Here D^α is the standard Riemann-Liouville fractional derivative of order α, f is a Carathéodory function and f(t,x,y,z) is singular at the value 0 of its arguments x,y,z.     

Information storage and retrieval — mathematical foundations II (combinatorial problems)

The following combinatorial problem is fundamental for the design of file organizations minimizing simultaneously the storage space and the access time: For a given family $\text{M}$ of subsets of a finite set X, find a partial function S:X→X (if there is one), such that every M?$\text{M}$ can be written as M=lcubx,S(x),?,S^|M|-1 (x)rcub for a suitable x?X. We study the class of families admitting such a function S, as well as some of its subclasses defined by imposing certain restrictions on S (these restrictions reflect the structure of the storage medium to be used for storing a file).     

Practical constructive schemes for deterministic shared-memory access

We present three explicit schemes for distributingM variables amongN memory modules, whereM=Θ(N ^1.5),M = Θ(N ²), andM=Θ(N ³), respectively. Each variable is replicated into a constant number of copies stored in distinct modules. We show thatN processors, directly accessing the memories through a complete interconnection, can read/write any set ofN variables in worst-case timeO (N ^1/3),O(N ^1/2), andO(N ^2/3), respectively for the three schemes. The access times for the last two schemes are optimal with respect to the particular redundancy values used by such schemes. The address computation can be carried out efficiently by each processor without recourse to a complete memory map and requiring onlyO(1) internal storage.     

Maximality and totality of stable functions in the category of stable bifinite domains

This paper studies maximality and totality of stable functions in the category of stable bifinite domains. We present three main results as follows,o

1. (1)every maximum-preserving function is a maximal element in the stable function spaces;
2. (2)a maximal stable function f : D → E is maximum-preserving if D is maximum-separable and E is completely separable; and
3. (3)a stable bifinite domain D is maximum-separable if and only if for any locally distributive stable bifinite domain E, each maximal stable function f : D → E is maximum-preserving.

     

Fully dynamic biconnectivity in graphs

We present an algorithm for maintaining the biconnected components of a graph during a sequence of edge insertions and deletions. It requires linear storage and preprocessing time. The amortized running time for insertions and for deletions isO(m ^2/3 ), wherem is the number of edges in the graph. Any query of the form ‘Are the verticesu andv biconnected?’ can be answered in timeO(1). This is the first sublinear algorithm for this problem. We can also output all articulation points separating any two vertices efficiently. If the input is a plane graph, the amortized running time for insertions and deletions drops toO(√n logn) and the query time isO(log² n), wheren is the number of vertices in the graph. The best previously known solution takes timeO(n ^2/3 ) per update or query.     

L 2-Error bounds for approximate solutions of elliptic partial differential equations with Dirichlet-boundary conditions

New a posteriori (computable) upper bounds for theL ₂-norms, both ofD(u?v) and ofu?v are proposed, whereu is the exact solution of the boundary value problem $$Au: = - D(pDu) + qu = f, x \in G and u = 0,x \in \partial G$$ andv any approximation of it (D is here the vector of partial derivatives with respect to the components ofx). It is shown that the new error bounds are better than the classical one, which is proportional to ‖Av?f‖, in many cases. This happens, e. g., ifq has some zero point inG, as in the case of a Poisson equation.