A　linear　difference　Hopfield　neural　network　which　has　the　function　of　iterative　learning　is　proposed　to　overcome　the　local　minimum　problem　of　its　energy　function.　Theoretical　analysis　shows　that　the　linear　Hopfield　neural　network　is　stable,　and　the　stable　state　makes　its　energy　function　reach　its　unique　minimum.　On　the　basis　of　the　relation　between　the　stability　of　the　linear　difference　Hopfield　network　and　its　energy　function＇s　convergence,　the　linear　Hopfield　network　is　applied　to　solve　linear　quadratic　optimization　control　problems　for　multivariable　time-varying　systems.　The　theoretical　design　method　of　linear　Hopfield　neural　network　shows　that　its　stable　outputs　are　the　desired　optimal　control　inputs.　The　simulation　results　are　in　accord　with　theoretical　analysis.