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We propose a novel recursive utility for controlling stochastic processes under risk and uncertainty. Our formulation uses a robustified Orlicz risk that can evaluate risk and uncertainty simultaneously. We focus on a control problem of a photovoltaic power generation system that supplies excess electricity to a secondary purpose for generating green hydrogen. The corresponding Hamilton-Jacobi-Bellman equation having a novel nonlinear term is then derived. Computational examples with the available data are finally presented, demonstrating that our methodology can be used for the photovoltaic power generation under different meteorological and operational conditions.

We formulate a new two-variable river environmental restoration problem based on jump stochastic differential equations (SDEs) governing the sediment storage and nuisance benthic algae population dynamics in a dam-downstream river. Controlling the dynamics is carried out through impulsive sediment replenishment with discrete and random observation/intervention to avoid sediment depletion and thick algae growth. We consider a cost-efficient management problem of the SDEs to achieve the objectives whose resolution reduces to solving a Hamilton-Jacobi-Bellman (HJB) equation. We also consider a Fokker-Planck (FP) equation governing the probability density function of the controlled dynamics. The HJB equation has a discontinuous solution, while the FP equation has a Dirac's delta along boundaries. We show that the value function, the optimized objective function, is governed by the HJB equation in the simplified case and further that a threshold-type control is optimal. We demonstrate that simple numerical schemes can handle these equations. Finally, we numerically analyze the optimal controls and the resulting probability density functions., Comment: Accepted manuscript

The objectives and contributions of this paper are mathematical and numerical analyses of a stochastic control problem of bounded population dynamics under ambiguity, an important but not well-studied problem, focusing on the optimality equation as a nonlinear degenerate parabolic partial integro-differential equation (PIDE). The ambiguity comes from lack of knowledge on the continuous and jump noises in the dynamics, and its optimization appears as nonlinear and nonlocal terms in the PIDE. Assuming a strong dynamic programming principle for continuous value functions, we characterize its solutions from both viscosity and distribution viewpoints. Numerical computation focusing on an ergodic case are presented as well to complement the mathematical analysis.

A new stochastic control problem of a dam-reservoir system installed in a river is analyzed both mathematically and numerically. Water balance dynamics of the reservoir are piece-wise deterministic and are driven by a stochastic regime-switching inflow process. The system is controlled to balance among the operation purpose and the internal and downstream environmental conditions. Finding the optimal operation policy of the system reduces to solving an optimality equation with a discontinuous Hamiltonian, which is a system of nonlinear degenerate parabolic (or hyperbolic) equations. We show that the optimality equation has at most one constrained viscosity solution and find the solution explicitly under certain conditions. The model is applied to numerical computation of the operation policy of an existing dam-reservoir system using a high-order finite difference scheme. The computational results can suggest how the operation policy should be adapted according to the environmental concerns of the river., Comment: This is a first revised form of the manuscript

Spin excitation in a nano-graphite ribbon with zigzag edges is investigated theoretically. Due to the strongly localized nature of the states near Fermi energy, the effective Hamiltonian for the low energy physics is given by Heisenberg Hamiltonian with the nearest neighbor exchange coupling. The action corresponding to the effective Hamiltonian is mapped to that of the O(3) nonlinear sigma model. It is shown that the spin excitation has a gap when the number of the zigzag lines is even, whereas the excitation becomes gapless in case of the odd number of the zigzag lines., Comment: 4 pages, 3 figures, submitted to J. Phys. Soc. Jpn

Tomonaga-Luttinger-liquid theory is formulated for metallic carbon nanotubes with open boundaries. Both cases of single- and multi-wall nanotubes are discussed. Based on this theory, spatial variation of the charge density from an edge is investigated with taking account of the shift of the chemical potential which expresses the carrier injection to the nanotube. The charge density has the spatially independent part and the oscillatory component. Roles of Coulomb interaction on the amplitude of the oscillation, the wavenumbers of it and the uniform component of the charge density are clarified., Comment: 14 pages, 4 figures, to be published in J. Phys. Soc. Jpn. Vol. 71 (2002) No.10

Magnetic properties in a charge ordered state are examined for the extended Hubbard model at half-filling. Magnetic excitations, magnetic susceptibilities and a nuclear spin relaxation rate are calculated with taking account of fluctuations around the mean-field solution. The relevance of the present results to the observation in the 1:1 organic conductors, (TTM-TTP)I$_3$, is discussed., Comment: 4 pages, 3 figures, to be published in J. Phys. Soc. Jpn. Vol.71 (2002) No.8

Based on the tight binding method with hopping integral between the nearest-neighbor atoms, an oscillator strength $\int_0^{\infty} \d \omega {\rm Re} \sigma (\omega)$ is discussed for armchair and metallic zigzag carbon nanotubes. The formulae of the oscillator strength are derived for both types of nanotubes and are compared with the result obtained by a linear chain model. In addition, the doping dependence is investigated in the absence of Coulomb interaction. It is shown that the oscillator strength of each carbon nanotube shows qualitatively the same doping dependence, but the fine structure is different due to it's own peculiar band structure. Some relations independent of the radius of the tube are derived, and a useful formula for determining the amount of doping is proposed., Comment: 4 pages, 4 figures, submitted to J. Phys. Soc. Jpn. at June 30, 2000

We examine effects of the next-nearest-neighbor repulsion on electronic states of a one-dimensional interacting electron system which consists of quarter-filled band and interactions of on-site and nearest-neighbor repulsion. We derive the effective Hamiltonian for the electrons around wave number $\pm \kf$ ($\kf$: Fermi wave number) and apply the renormalization group method to the bosonized Hamiltonian. It is shown that the next-nearest-neighbor repulsion makes $4\kf$-charge ordering unstable and suppresses the spin fluctuation. Further the excitation gaps and spin susceptibility are also evaluated., Comment: 19 pages, 8 figures, submitted to J. Phys. Soc. Jpn

Single-wall carbon nanotubes are almost ideal systems for the investigation of exotic many-body effects due to non-Fermi liquid behavior of interacting electrons in one dimension. Recent theoretical and experimental results are reviewed with a focus on electron correlations. Starting from a microscopic lattice model we derive an effective phase Hamiltonian for conducting single-wall nanotubes with arbitrary chirality. The parameters of the Hamiltonian show very weak dependence on the chiral angle, which makes the low-energy physics of conducting nanotubes universal. The temperature-dependent resistivity and frequency-dependent optical conductivity of nanotubes with impurities are evaluated within the Luttinger-like model. Localization effects are studied. In particular, we found that intra-valley and inter-valley electron scattering can not coexist at low energies. Low-energy properties of clean nanotubes are studied beyond the Luttinger liquid approximation. The strongest Mott-like electron instability occurs at half filling. In the Mott insulating phase electrons at different atomic sublattices form characteristic bound states. The energy gaps of $0.01-0.1$ eV occur in all modes of elementary excitations. We finally discuss observability of the Mott insulating phase in transport experiments. The accent is made on the charge transfer from external electrodes which results in a deviation of the electron density from half-filling., Comment: 16 pages, 4 figures, Latex style files included, Submitted to Proceedings of Heraeus Workshop WEH219, Hamburg 1999

Properties of fluctuations in two chains of Tomonaga-Luttinger liquids coupled by the interchain hopping have been studied by calculating retarded response functions $\chi^R_{\rho} (q_x,q_y;\omega)$ for charge and $\chi^R_{\p} (q_x,q_y;\omega)$ for spin where $q_x$ and $q_y (=0$ or $\pi$) denote the longitudinal and transverse wave vector, respectively, and $\omega$ is the frequency. We have found the notable fact that the repulsive intrachain interaction results in the clear enhancement of ${\rm{Im}}\chi^R_\p (q_x,\pi;\omega)$ and the suppression of ${\rm{Im}}\chi^R_\rho (q_x,\pi;\omega)$ at low energies. This result indicates the importance of the dynamical effect by the spin fluctuation with $q_y = \pi$ and small $\omega$, which has a possibility to give rise to the attractive interaction for the electron pairing., Comment: 8 pages, 5 figures, to be published in Physica C

The persistent current of interacting electrons in toroidal single-wall carbon nanotubes is evaluated within Haldane's concept of topological excitations. The overall pattern of the persistent current corresponds to the constant interaction model, whereas the fine structure stems from the electronic exchange correlations., Comment: 4 pages, 3 figures

The phase Hamiltonian of armchair carbon nanotubes at half-filling and away from it is derived from the microscopic lattice model by taking the long range Coulomb interaction into account. We investigate the low energy properties of the system using the renormalization group method. At half-filling, the ground state is a Mott insulator with spin gap, in which the bound states of electrons at different atomic sublattices are formed. The difference from the recent results [Phys. Rev. Lett. 79, 5082 (1997)] away half-filling is clarified., Comment: 4 pages, 1 figure, Revtex