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Asymptotically minimax nonparametric estimation of a regression function observed in white Gaussian noise over a bounded interval is considered, with respect to a L2-loss function. The unknown function f is assumed to be m times differentiable except for an unknown although finite number of jumps, with piecewise mth derivative bounded in L2 norm. An estimator is constructed, attaining the same optimal risk bound, known as Pinsker's constant, as in the case of smooth functions (without jumps).

The problem considered here is an infinite stage inventory decision process in which the penalty cost for failing to have in stock sufficient items to satisfy demand at the next stage includes a fixed administrative cost (red tape) and a cost proportional to the deficiency in stock. It is shown here that when the demand follows a Weibull distribution, optimal constant stock level solution exists. A simple computer program is given to calculate this optimal solution.

This paper considers the sequencing of n jobs in a single-machine shop in which each job is assigned a constant flow allowance. The objective is to determine the optimal value of the flow allowance and the optimal job sequence to minimize a cost function based on the flow allowance and the job earliness and tardiness values. We first propose a linear programming (LP) formulation of the problem. We then derive the optimal constant flow allowance via considering the LP dual problem and show that it is independent of the job sequence. After the theoretical treatment, a numerical example is presented for discussion.

This paper considers the problem of determining the optimal constant flow allowance for the CON due-date assignment method. A linear programming formulation of the problem is proposed and the optimal due-date is obtained by solving the LP dual problem.

The smallest nonzero Stekloff eigenvalue $\xi _2 $ satisfying $\Delta ^2 w = 0$ in R, ${{\partial w} / {\partial n}} = {{\partial \Delta w} / {\partial n}} + \xi w = 0$ on $\partial R$, is an optimal constant in the a priori inequality used to approximate, solutions of the Neumann problem for Poisson’s equation. Tight bounds are given for $\xi _2 $ on rectangles by the method of a posteriors–a priori inequalities.

This paper considers the problem of optimal constant due-date assignment and sequencing of jobs in a single-machine shop. We formulate the problem as a general constrained optimization problem and apply the Kuhn-Tucker conditions to find the optimal solution which is shown to be independent of the job sequence.

We consider a class of ecosystems subjected to optimal constant harvesting effort and continual, unpredictable, but bounded disturbances. If certain conditions are satisfied, a state feedback control can be used to guarantee uniform ultimate bounded-ness of every system response within an arbitrarily small neighbourhood of the unique non-trivial equilibrium state of the undisturbed exploited ecosystem. The result is demonstrated with the aid of some examples

This paper develops general overtaking techniques for studying the asymptotic properties of portfolio policies optimal with respect to a terminal utility valuation. For a restricted class of utility functions the sequence of optimal constant (non-revised) portfolio policies formed as the horizon recedes into the future is shown to converge. Furthermore, for utility functions unbounded above and below, this turnpike policy need not be the policy associated with the minimal constant relative risk aversion function that bounds the valuation function from above. Finally, an analogy between the portfolio turnpike problem and the turnpike problem of growth theory is studied.

A dual version of the Levine-Athans algorithm for solving the optimal constant output feedback gain for continuous-time systems is studied. Global convergence of the procedure to a stationary point of the loss function is proved. A global convergence property of the Levine-Athans algorithm is also obtained.

A model for cyclical pressure freeze drying has been solved numerically and the results used to evaluate the effectiveness of the technique when compar However, all the cyclical policies tried were inferior (although only slightly in some cases) to a near optimal constant pressure policy developed prev

We show that for every Banach lattice E E failing RNP and not containing c 0 {c_0} (resp. containing c 0 {c_0} ) and for every ε > 0 \varepsilon > 0 there exists a solid convex closed subset D D of the unit ball of E E , such that \[ dist ⁡ [ ext ⁡ ( D ¯ σ ( E ∗ ∗ , E ∗ ) ) , E ] > 1 − ε ( resp . > 1 2 − ε ) \operatorname {dist} [\operatorname {ext} ({\overline D ^{\sigma ({E^{ * * }},{E^ * })}}),E] > 1 - \varepsilon (\operatorname {resp} . > \tfrac {1}{2} - \varepsilon ) \] and such that every slice of D D has diameter bigger than 2 − ε 2 - \varepsilon . We also prove that these results are optimal. We apply them to construct rough lattice norms with almost optimal constant on non-Asplund Banach lattices.

The optimal constant gain output feedback control of linear time-invariant discrete systems with respect to an averaged performance criterion is discussed. The algebraic conditions necessary for minimizing the averaged performance cost are derived and a method for computing the constant optimal feedback gain is presented.

Necessary and sufficient globally optimal conditions—a matrix equation and a matrix inequality—are given for the existence of the optimal constant output feedback gain. Furthermore, it is shown that if the optimal output law L0 exists, it must be a solution derived from the corresponding optimal state-variable feedback problem, that is L0C = K0, where K0 is the optimal state-variable feedback law. An example is given to show that a globally optimal output law may not be found even if the system is stabilizable by output feedback. These results are helpful for the understanding of the fundamental problems of output feedback and give the reasons for suboptimal approaches as adopted by Levine and Athans (1970) and other authors.

In this paper we determine the maximum cardinality of a packing of K4's into Kn, that is, construct optimal constant weight codes with weight 4 and minimum distance 6.

One of the basic problems in Periodic Optimization is to ascertain by simple calculations whether the optimal constant control of a given plant can be improved by cycling or not. In this direction, one of the most powerful results is a frequency domain condition usually referred to as Π-criterion. The main limitation to the use of such a criterion is the assumption that the optimal constant control be an interior point of the admissible control set. In this paper, an extension of the Π-criterion to the case of boundary optimal constant control is presented and a particular case discussed in which the condition above can be given the useful form of a circle criterion.

Suppose prices p are adjusted in accordance with excess demand z by Δp = k 1 z . We determine the optimal constant k 1 given that demand is the sum of a linear function of price and a Markov disturbance. Supply is fixed.

This paper considers an application of control theory to management decision problems. The formulation of optimal periodic control is employed in studying the so-called intertemporal price discrimination problem. Previous static analyses of this problem have shown that the policy of switching between two certain price levels is more profitable than the optimal constant price. In the present paper the dynamics of the demand is modelled explicitly. Using this formulation the optimal dynamic solution to this problem can be found. A numerical example with three consumer groups is considered. The solution to this example shows that the optimal price patterns need not necessarily consist of only two price levels.

The problem is discussed of establishing whether the optimal constant control of a given plant (with respect to a specified performance index and subject to equality and/or inequality constraints) can be improved by cycling. A frequency domain approach is followed in order to establish a general condition, which, in the particular case of single-input systems, admits a simple geometric interpretation. Connections with high frequency and slowly varying periodic control problems are also pointed out.

We establish a class of sharp logarithmic estimates for the Beurling-Ahlfors transform B on the complex plane. For any K > 0 we determine the optimal constant \({L = L(K) \in (0, \infty]}\) such that the following holds. If \({F : \mathbb{C} \rightarrow \mathbb{C}}\) is a radial function, then for any R > 0, $$\frac{1}{|\mathcal{B}(0, R)|} \int_{\mathcal{B}(0, R)} |BF(z)| dz \leq \frac{K}{|\mathcal{B}(0, R)|} \int_{\mathcal{B}(0, R)} \Psi(|F(z)|) dz + L(K),$$ where Ψ(t) = (t + 1) log(t + 1) – t and \({\mathcal{B}(0, R) \subset \mathbb{C}}\) denotes the ball of center 0 and radius R. A related result in higher dimensions is also established. The proof rests on probabilistic methods and exploits a certain sharp inequality for martingales.

For any 1 < p < ∞ and any \({X, Y\in \mathbb{R}}\) satisfying \({|X|\leq Y}\) , we determine the optimal constant Cp(X,Y) such that the following holds. If F is a holomorphic function on the unit disc satisfying ReF(0) = X and \({||{\rm Re}F||_{L^{p}(\mathbb{T})}=Y}\) , then $$||F||_{L^p(\mathbb{T})}\geq C_p(X,Y).$$ This can be regarded as a reverse version of the classical estimates of Riesz and Essen. The proof rests on the exploitation of certain families of special subharmonic functions on the plane.

A design problem of optimal regulators for discrete-time linear multi-input systems is investigated for the prespecified closed-loop eigenvalues. Then the design freedom, which remains after the prespecified closed-loop eigenvalues are assigned, is used to minimize a time-weighted performance index. Necessary conditions for the optimality of the controller that satisfies the prespecified closed-loop eigenvalues are derived and an algorithm for computing the optimal constant feedback gain is presented. >

The design problem of optimal output feedback controller for discrete-time linear time-invariant systems is investigated with a class of time-multiplied quadratic performance indexes. Necessary conditions for optimality are derived. It is shown via example that the design method using such a time-multiplied quadratic performance index improves the transient response of the system.

Recently Bittanti et al. [1] analyzed the problem of establishing whether the optimal constant control of a system can be improved by cycling. Here the question is raised how their results could be used to approach the optimal periodic operation as closely as possible. A partial answer may be provided by classical perturbation techniques.

Let 1 Ω r ≔ {x ∈ R N : 0 . We prove that there exists q, p ∗ = Np (N − p) such that the extremal for the optimal constant C≔ C(p, q, Ω r ) of the embedding W 0 1,p (Ω r ) ↪↪ L q (Ω r ) is not a radial function (and hence, not unique). As a consequence of this fact, we obtain a nonuniqueness result for −Δ p u = |u| q−2 u, u > 0 in Ω r , u = 0 on ∂Ω r , ‖∇u‖ p = C q (p−q ) .

The stationary and dynamic behaviour of multi-motor drives in processing machines with continuous moving webs will be investigated. The influence of the web forces is neglected up to now normally during the optimization of the control. Poor dynamic behaviour is the result of this simplification. To improve the dynamic behaviour of the controlled system, new concepts with adaptive state space controllers and observers are considered. As a first step the decoupling method for the multi-motor drive-system with decentralized observers will be investigated. Then the optimal constant output feedback will be applied and at last an adaptive state space control for axial winders is described. The results, obtained from these new approaches will be compared to the results of conventional approaches. Furthermore practical experiments show the advantages of the new concepts.

The problem of orbit control for spacecraft on elliptical orbits is analysed and an approach to the design of optimal constant gain controllers for the periodic dynamics of relative motion is proposed. In particular, it is shown how the proposed approach can guarantee closed loop stability and optimal performance both in the case of circular and elliptical orbits.

In this paper, we prove the estimate $$\left| {deg g} \right| \leqslant C\mathop {\smallint _{\mathbb{S}^N } \smallint _{\mathbb{S}^N } }\limits_{\left| {g(x) - g(y)} \right| > \delta } \frac{1}{{\left| {x - y} \right|^{2N} }}dxdy, \forall g \in C(\mathbb{S}^N ,\mathbb{S}^N ),$$ , for every δ ∈ (0, lN), where C = C(N) is a positive constant depending only on N and $$\ell _N = \sqrt {2 + 2/(N + 1)} $$ . We show that the constant lN in this estimate is optimal. We also present a class of maps from $$\mathbb{S}^N $$ into $$\mathbb{S}^N $$ , strictly larger than $$C(\mathbb{S}^N ,\mathbb{S}^N )$$ , on which we can define the notion of degree and for which the previous inequality still holds.

A simple and direct derivation of necessary conditions for optimality of output feedback gains for a linear time-invariant system is presented. The dependence of the cost on the initial state x(0) is eliminated by the well-known technique of averaging the initial state over the surface of the n -dimensional unit sphere.

A concise derivation of the necessary conditions that must be satisfied by optimal constant limited state feedback gains is given. The approach taken to obtain these conditions is applicable to other limited state feedback controller design problems as well.

We show that a Hardy-Rellich inequality with optimal constants on a bounded domain can be refined by adding remainder terms. The procedure is based on decomposition into spherical harmonics.