On the null-controllability of nonlinear delay systems with restrained controls

Sufficient conditions are developed for the null-controllability of the nonlinear delay process (1) $\text{x}\text{?}\text{(t) = L(t, x}{}_{\mathrm{t}}\text{) + B(t) u(t) + f(t, x}{}_{\mathrm{t}}\text{, u(t))}$ when the values of the control functions u lie in an m-dimensional unit cube C^m of E^m. Conditions are placed on f which guarantee that if the uncontrolled system $\text{x}\text{?}\text{(t) = L(t, x}{}_{\mathrm{t}}\text{)}$ is uniformly asymptotically stable and if the linear control system x(t) = L(t, x_t) + B(t) u(t) is proper, then (1) is null-controllable.     

Extensions of unbounded 1-derivations in UHF C1-algebras

We consider unbounded ¹-derivations δ in UHF-C¹-algebras A=(∪^∞_n=1A_n)^?) with dense domain. If ?_n:A→A_n denotes the conditional expectations onto the finite type I factors $\text{A}$_n, then we introduce a weak-commutativity condition for δ and the sequence (?_n). As a consequence of this condition on δ we establish the existence of an extension derivation δ′ which is the infinitesimal generator of a strongly continuous one-parameter group, α: $\text{R}$ → Aut($\text{A}$), of ¹-automorphisms, i.e., $\text{δ′(x) =}\text{(d}\text{dt)}\text{α}{}_{\mathrm{t}}\text{(x)¦}{}_{\mathrm{t\; =\; 0}}$ for x?D(δ′). Special properties of α (alias δ′) are considered. We show that AF-algebras are associated to proper restrictions δ of derivations δ′ of product type. We then turn to the extendability problem for quasifree derivations in the CAR-algebra. There, extensions δ′ are calculated which generate strongly continuous semigroups of ¹-homomorphisms. These semigroups do not extend to one-parameter groups unless the implementing symmetric operator in one-particle space is already self-adjoint.     

Periodic forcing of solutions of a boundary value problem for a second-order differential equation in Hilbert space

We show that if u is a bounded solution on $\text{R}$⁺ of u″(t) ?Au(t) + f(t), where A is a maximal monotone operator on a real Hilbert space H and f∈L_loc²($\text{R}$⁺;H) is periodic, then there exists a periodic solution ω of the differential equation such that u(t) ? ω(t)   0 and u′(t) ? ω′(t) → 0 as t → ∞. We also show that the two-point boundary value problem for this equation has a unique solution for boundary values in $\text{D(A)}$ and that a smoothing effect takes place.     

Boundary layer methods for certain nonlinear singularly perturbed optimal control problems

We shall examine the control problem consisting of the system $\text{dx}\text{dt}\text{= f}{}_1\text{(x, z, u, t, ?)}$$\text{?(}\text{dz}\text{dt}\text{) = f}{}_2\text{(x, z, u, t, ?)}$ on the interval 0 ? t ? 1 with the initial values x(0, ?) and z(0, ?) prescribed, where the cost functional J(?) = π(x(1, ?), z(1, ?), ?) + ∝₀¹V(x(t, ?), z(t, ?), u(t, ?), t, ?) dt is to be minimized. We shall restrict attention to the special problem where the f_i's are linear in z and u, V is quadratic in z and independent of z when ? = 0, π and V are positive semidefinite functions of x and z, and V is a positive definite function of u. Under appropriate conditions, we shall obtain an asymptotic solution of the problem valid as the small parameter ? tends to zero. The techniques of constructing such asymptotic expansions will be stressed.     

Existence and comparison theorems for nonlinear diffusion systems

In this paper we prove existence, uniqueness, and regularity results for systems of nonlinear second order parabolic equations with boundary conditions of the Dirichlet, Neumann, and regular oblique derivative types. Let K(t) consist of all functions (v¹(x), v²(x),…, v^m(x)) from $\text{Ω ? R}{}^{\mathrm{n}}$ into R^m which satisfy ψⁱ(x, t) ? vⁱ(x) ? θⁱ(x, t) for all $\text{x ? Ω}\text{and}\text{1 ? i ? m}$, where ψⁱand θⁱ are extended real-valued functions on $\text{}\text{?}\text{gW × [0, T)}$. We find conditions which will ensure that a solution U(x, t) ≡ (u¹(x, t), u²(x, t),…, u^m(x, t)) which satisfies U(x, 0) ?K(0) will also satisfy U(x, t) ?K(t) for all 0 ? t < T. This result, which has some similarity to the Gronwall Inequality, is then used to prove a global existence theorem.     

On a generalization of Minkowski's convex body theorem

Given a lattice $\text{Λ ? R}{}^{\mathrm{n}}$ and a bounded function g(x), x ∈ Rⁿ, vanishing outside of a bounded set, the functions ?(x)$\text{g}\text{?}\text{(x)?}$max_u∈Λg(u +x), ?(x)?Σ_{u∈Λ g(u +x)}, and ?₊(x)?Σ_u∈Λ max_{v∈Λ min {g(v + x); g(u + v + x)}} are defined and periodic mod Λ on Rⁿ. In the paper we prove that ?(x) + ?₊(x) ? 2?(x) ≥ ?(x) + h?₊(x) ? 2?(x) holds for all x ∈ Rⁿ, where h(x) is any “truncation” of g by a constant c ≥ 0, i.e., any function of the form h(x)?g(x) if g(x) ≤ c and h(x)?c if g(x) > c. This inequality easily implies some known estimations in the geometry of numbers due to Rado [1] and Cassels [2]. Moreover, some sharper and more general results are also derived from it. In the paper another inequality of a similar type is also proved.     

Asymptotic properties of general symmetric hyperbolic systems

Results on partition of energy and on energy decay are derived for solutions of the Cauchy problem $\text{?u}\text{?t}\text{+ ∑}{}_{\mathrm{j\; =\; 1}}{}^{\mathrm{n}}\text{A}{}_{\mathrm{j}}\text{?u}\text{?x}{}_{\mathrm{j}}\text{= 0, u(0, x) = ?(x)}$. Here the A_j's are constant, k × k Hermitian matrices, x = (x₁,…, x_n), t represents time, and u = u(t, x) is a k-vector. It is shown that the energy of Mu approaches a limit $\text{E}{}_{\mathrm{M}}\text{(?)}\text{as}\text{¦ t ¦ → ∞}$, where M is an arbitrary matrix; that there exists a sufficiently large subspace of data ?, which is invariant under the solution group U₀(t) and such that $\text{U}{}_0\text{(t)? = 0}\text{for}\text{¦ x ¦ ? a ¦ t ¦ ? R, a}\text{and}\text{R}$ depending on ? and that the local energy of nonstatic solutions decays as $\text{¦ t ¦ → ∞}$. More refined results on energy decay are also given and the existence of wave operators is established, considering a perturbed equation $\text{E(x)}\text{?u}\text{?t}\text{+ ∑}{}_{\mathrm{j\; =\; 1}}{}^{\mathrm{n}}\text{A}{}_{\mathrm{j}}\text{?u}\text{?x}{}_{\mathrm{j}}\text{= 0,}\text{where}\text{¦ E(x) ? I ¦ = O(¦ x ¦}{}^{\mathrm{?1\; ?\; ?}}\text{)}$ at infinity.     

Construction of translation planes from t-spread sets

A t-spread set [1] is a set $\text{C}$ of (t + 1) × (t + 1) matrices over GF(q) such that ∥C∥ = q^t+1, 0 ? C, I?C, and det(X ? Y) ≠ 0 if X and Y are distinct elements of $\text{C}$. The amount of computation involved in constructing t-spread sets is considerable, and the following construction technique reduces somewhat this computation. Construction: Let $\text{G}$ be a subgroup of GL(t + 1, q), (the non-singular (t + 1) × (t + 1) matrices over GF(q)), such that ∥G∥|a^t+1, and det (G ? H) ≠ 0 if G and H are distinct elements of $\text{G}$. Let A₁, A₂, …, A_n?GL(t + 1, q) such that det(A_i ? G) ≠ 0 for i = 1, …, n and all G?G, and det(A_i ? A_jG) ≠ 0 for i > j and all G?G. Let $\text{C = \&{0\&} ∪ G ∪ A}{}_1\text{G ∪ … ∪ A}{}_{\mathrm{n}}\text{G}$, and ∥C∥ = q^t+1. Then $\text{C}$ is a t-spread set. A t-spread set can be used to define a left V ? W system over V(t + 1, q) as follows: x + y is the vector sum; let e?V(t + 1, q), then xoy = yM(x) where M(x) is the unique element of $\text{C}$ with x = eM(x). Theorem: Let$\text{C}$be a t-spread set and F the associatedV ? Wsystem; the left nucleus = {y | CM(y) = C}, and the middle nucleus = }y | M(y)C = C}. Theorem: For$\text{C}$constructed as aboveG ? {M(x) | x?N_λ}. This construction technique has been applied to construct a V ? W system of order 25 with ∥N_λ∥ = 6, and ∥N_μ∥ = 4. This system coordinatizes a new projective plane.     

An abstract averaging theorem

For each t ? 0, let A(t) generate a contraction semigroup on a Banach space L. Suppose the solution of u_t = ?A(t)u is given by an evolution operator V_?(t, s). Conditions are given under which $\text{V}{}_{\mathrm{?}}\text{(}\text{(t+s)}\text{?}\text{,}\text{s}\text{?}\text{)}$ converges strongly as ? → 0 to a semigroup T(t) generated by the closure of $\text{A}\text{?}\text{f ≡}\text{lim}{}_{\mathrm{T\to \infty }}\text{(}\text{1}\text{T}\text{) ∝}{}_0{}^{\mathrm{T}}\text{A(t)f dt}$.This result is applied to the following situation: Let B generate a contraction group S(t) and the closure of ?A + B generate a contraction semigroup S_?(t). Conditions are given under which $\text{S(?}\text{t}\text{?}\text{) S}{}_{\mathrm{?}}\text{(}\text{t}\text{?}\text{)}$ converges strongly to a semigroup generated by the closure of $\text{A}\text{?}\text{f ≡}\text{lim}{}_{\mathrm{T\to \infty }}\text{(}\text{1}\text{T}\text{) ∝ S(?t) AS(t)f dt}$. This work was motivated by and generalizes a result of Pinsky and Ellis for the linearized Boltzmann Equation.     

The operator-valued Feynman-Kac formula with noncommutative operators

Let X(t) be a right-continuous Markov process with state space E whose expectation semigroup S(t), given by S(t) φ(x) = E_x[φ(X(t))] for functions φ mapping E into a Banach space L, has the infinitesimal generator A. For each x?E, let V(x) generate a strongly continuous semigroup T_x(t) on L. An operator-valued Feynman-Kac formula is developed and solutions of the initial value problem $\text{?u}\text{?t}\text{= Au + V(x)u, u(0) = φ}$ are obtained. Fewer conditions are assumed than in known results; in particular, the semigroups {T_x(t)} need not commute, nor must they be contractions. Evolution equation theory is used to develop a multiplicative operative functional and the corresponding expectation semigroup has the infinitesimal generator A + V(x) on a restriction of the domain of A.     

Nonlinear Volterra equations on a Hilbert space

We consider equations of the form, u(t) = ? ∝₀^tA(t ? τ)g(u(τ)) dτ + ?(t) (I) on a Hubert space $\text{H}$. A(t) is a family of bounded, linear operators on $\text{H}$ while g is a transformation on $\text{D}$_g ?$\text{H}$ which can be nonlinear and unbounded. We give conditions on A and g which yield stability and asymptotic stability of solutions of (I). It is shown, in particular, that linear combinations with positive coefficients of the operators e^Mt and ?e^Mtsin Mt where M is a bounded, negative self-adjoint operator on $\text{H}$ satisfy these conditions. This is shown to yield stability results for differential equations of the form, $\text{Q (}\text{d}\text{dt}\text{) u = ? P (}\text{d}\text{dt}\text{) g(u(t)) + χ(t)}$, on $\text{H}$.     

Approximate solutions of functional differential equations

The author discusses the best approximate solution of the functional differential equation x′(t) = F(t, x(t), x(h(t))), 0 < t < l satisfying the initial condition x(0) = x₀, where x(t) is an n-dimensional real vector. He shows that, under certain conditions, the above initial value problem has a unique solution y(t) and a unique best approximate solution $\text{p}\text{?}{}_{\mathrm{k}}\text{(t)}$ of degree k (cf. [1]) for a given positive integer k. Furthermore, $\text{sup}{}_{\mathrm{0?t?l}}\text{¦}\text{p}\text{?}{}_{\mathrm{k}}\text{(t) ? y(t)¦ → 0}\text{as}\text{k → ∞}$, where ¦ · ¦ is any norm in Rⁿ.     

Potential theory associated with Uhlenbeck-Ornstein process

Some parallel results of Gross' paper (Potential theory on Hilbert space, J. Functional Analysis1 (1967), 123–181) are obtained for Uhlenbeck-Ornstein process U(t) in an abstract Wiener space (H, B, i). Generalized number operator $\text{N}$ is defined by $\text{N}$f(x) = ?lim_∈←0{E[f($\text{U}$(τ^∈_ξ))] ? f(x)}/E[τ^∈_ξ, where τ_x^? is the first exit time of U(t) starting at x from the ball of radius ? with center x. It is shown that $\text{N}$f(x) = ?trace D²f(x)+〈Df(x),x〉 for a large class of functions f. Let r_t(x, dy) be the transition probabilities of U(t). The λ-potential G_λf, λ > 0, and normalized potential Rf of f are defined by G_λf(X) = ∫₀^∞e^?λtr_tf(x) dt and Rf(x) = ∫₀^∞ [r_tf(x) ? r_tf(0)] dt. It is shown that if f is a bounded Lip-1 function then trace D²G_λf(x) ? 〈DG_λf(x), x〉 = ?f(x) + λG_λf(x) and trace D²Rf(x) ? 〈DRf(x), x〉 = ?f(x) + ∫_Bf(y)p₁(dy), where p₁ is the Wiener measure in B with parameter 1. Some approximation theorems are also proved.     

Threshold phenomena for a reaction-diffusion system

We consider the pure initial value problem for the system of equations $\text{ν}{}_{\mathrm{t}}\text{= ν}{}_{\mathrm{xx}}\text{+ ?(ν) ? w, w}{}_{\mathrm{t}}\text{= ε(ν ? γw), ε, γ ? 0}$, the initial data being (ν(x, 0), w(x, 0)) = (?(x), 0). Here $\text{?(v) = ?v + H(v ? a)}$, where H is the Heaviside step function and $\text{a ? (0,}\text{1}\text{2}\text{)}$. This system is of the FitzHugh-Nagumo type and has several applications including nerve conduction and distributed chemical/ biochemical systems. It is demonstrated that this system exhibits a threshold phenomenon. This is done by considering the curve s(t) defined by s(t) = sup{x: v(x, t) = a}. The initial datum, ?(x), is said to be superthreshold if lim_{t→∞ s(t) = ∞}. It is proven that the initial datum is superthreshold if ?(x) > a on a sufficiently long interval, ?(x) is sufficiently smooth, and ?(x) decays sufficiently fast to zero as $\text{¦x¦ → ∞}$.     

Générateurs et semi-groupes dans les espaces de Banach uniformement lisses

Let C be a closed convex subset of a uniformly smooth Banach space. Let S(t) : C → C be a semigroup of type ω. Then the generator A₀ of S(t) has a dense domain in C. Moreover there is is an operator A such that: (i) A₀ ? A and $\text{D(A)}\text{= C, (ii) A + gwI}$ accretive, (iii) R(I + λA) ? C for λ > 0 and ωλ < 1, (iv) $\text{S(t)x =}\text{lim}{}_{\mathrm{n\; \to \; \infty }}\text{(I + (}\text{t}\text{n}\text{)A)}{}^{\mathrm{?n}}\text{x}$ for every x?C.     

A Wiener test for integrals of brownian motion and the existence of smooth curves in nowhere dense sets

Let J_ω^x(t) = x + ∝₀^tb_ω(s) ds, where b_ω is planar Brownian motion starting at 0. A Wiener-type criterion is proved for the process J_ω^x(t): Let K be a compact plane set and let x?K. Then if ∑ 2ⁿM₁(A_n(x)?K) < ∞ (where $\text{A}{}_{\mathrm{n}}\text{(x) = {2}{}^{\mathrm{?n?1}}\text{? ¦ z ? x ¦ ? 2}{}^{\mathrm{?n}}\text{}}$ and M₁ denotes one-dimensional Hausdorff content), the process J_ω^x(t) stays within K for a positive period of time t, a.s. In particular, this applies to almost all x with respect to area in the nowhere dense “Swiss Cheese” sets. The method is based on general potential theory for Markov processes.     

Entrainment of frequency in evolution equations

The existence of periodic solutions near resonance is discussed using elementary methods for the evolution equation ·u = Au + ?f(t, u) when the linear problem is totally degenerate (e^2πA = I) and the period of f is entrained with ? (T = 2π(1 + ?μ)). The approach is to solve the periodicity equation u(T,p,?) = p for an element p(?) in D, the domain of A, as a perturbation from an approximate solution p₀. p₀ is a solution of the nonlinear boundary value problem 2πμAp + ∝₀^2πe^?Asf(s, e^Asp) ds = 0 obtained from the periodicity equation by dividing by ?, applying the entrainment assumption, and letting ? → 0. Once p₀ is known, the conventional inverse function theorem is applied in a slightly unconventional manner. Two particular cases where results are obtained are u_t = u_x + ?{g(u) ? h(t, x)} with g strongly monotone and

$\text{d}\text{dt}\text{v}\text{w}\text{=}\text{0}\text{d}\text{dx}\text{d}\text{dx}\text{0}\text{v}\text{w}\text{+ ?}\text{v}{}^3\text{h(t,x)}$

, where in both cases D is a certain class of 2π-periodic functions of x.     

Estimates of derivatives of random functions I

If one looks for an optimal (by criterion of minimal variance) linear estimate of s′(t) from the observation of u(t) = s(t) + n(t), where n(t) is noise and s(t) is useful signal, then one can derive an integral equation for the weight function of optimal estimate. This integral equation is often difficult to solve and, even if one can solve it, it is difficult to construct the corresponding filter. In this paper an optimal estimate of s′ on a subset of all linear estimates in sought and it is shown that this quasioptimal estimate is easy to calculate, the corresponding filter is easy to construct, and the error of this estimate differs little from the error of optimal estimates. It is also shown that among all estimates (linear and nonlinear) of s′ for ∥n∥ ? δ and ∥s″∥ ? M the best estimate is given by Δ_hu = (2h)^?1 [u(t + h) ? u(t ? h)] with $\text{h = (}\text{2δ}\text{M}\text{)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$.     

More coverings by rook domains

The set V_kⁿ of all n-tuples (x₁, x₂,…, x_n) with x_i?, $\text{Z}$_k is considered. The problem treated in this paper is determining σ(n, k), the minimum size of a set W ? V_kⁿ such that for each x in V_kⁿ, there is an element in W that differs from x in at most one coordinate. By using a new constructive method, it is shown that σ(n, p) ? (p ? t + 1)p^n?r, where p is a prime and $\text{n = 1 +}\text{t(p}{}^{\mathrm{r?1}}\text{? 1)}\text{(p ? 1)}$ for some integers t and r. The same method also gives σ(7, 3) ? 216. Another construction gives the inequality σ(n, kt) ? σ(n, k)t^n?1 which implies that σ(q + 1, qt) = q^q?1t^q when q is a prime power. By proving another inequality σ(np + 1, p) ? σ(n, p)p^n(p?1), σ(10, 3) ? 5 · 3⁶ and σ(16, 5) ? 13 · 5¹² are obtained.     

Adiabatic invariants for linear Hamiltonian systems

n independent adiabatic invariants in involution are found for a slowly varying Hamiltonian system of order 2n × 2n. The Hamiltonian system considered is $\text{?}\text{u}\text{?}\text{= A(t)u}\text{as}\text{? → 0}{}^{\mathrm{+}}$, where A(t) is a 2n × 2n real matrix with distinct, pure imaginary eigen values for each t? [?∞, ∞], and $\text{d}{}^{\mathrm{(j)}}\text{A}\text{dt}{}^{\mathrm{(j)}}\text{? L}{}_{\mathrm{j}}\text{(?∞, ∞)}$, for all j > 0. The adiabatic invariants I_s(u, t), s = 1,…, n are expressed in terms of the eigen vectors of A(t). Approximate solutions for the system to arbitrary order of ? are obtained uniformly for t? [?∞, ∞].