您需要在对象中传递引用。我更改了mindate函数的名称，使其更加明确

function getMinDateFromEndDate(date) { //pass in the relative end date reference
    var oldDay = new Date(date);
    oldDay.setDate(oldDay.getDate() - 365);
    return getUTCDate(oldDay);
};

let endDate = getUTCDate(new Date('2021-01-03')); // set the arbitrary end date
let startDate = getMinDateFromEndDate(endDate)

$scope.logVariables = {
    startDate: startDate,
    endDate: endDate
}

$scope.dateOptions = {
    formatYear: "yyyy",
    minDate: startDate,
    maxDate: endDate,
    startingDay: 1
};

因为您使用的是utc时间，所以我编写了这个片段，展示了转换为utc的方法，以及如何使用utc设置最小和最大日期

function getUTCDate(date) {
   var now_utc = Date.UTC(date.getUTCFullYear(), date.getUTCMonth(), date.getUTCDate(), date.getUTCHours(), date.getUTCMinutes(), date.getUTCSeconds());
   return new Date(now_utc);
}

angular.module('dateExample', ['ngSanitize', 'ui.bootstrap'])
  .controller('ExampleController', ['$scope', function($scope) {

    function getUTCDate(date) {
      var now_utc = Date.UTC(date.getUTCFullYear(), date.getUTCMonth(), date.getUTCDate(),
        date.getUTCHours(), date.getUTCMinutes(), date.getUTCSeconds());
      return new Date(now_utc);
    }

    function getMinDateFromEndDate(date) {
      var oldDay = new Date(date);
      oldDay.setDate(oldDay.getDate() - 365);
      return getUTCDate(oldDay);
    };

    $scope.popup1 = {
      opened: false
    };
    $scope.popup2 = {
      opened: false
    };
    $scope.open1 = function() {
      $scope.popup1.opened = true;
    };

    $scope.open2 = function() {
      $scope.popup2.opened = true;
    };

    let endDate = getUTCDate(new Date('2021-01-03'));
    let startDate = getMinDateFromEndDate(endDate)
    $scope.logVariables = {
      startDate: startDate,
      endDate: endDate
    }
    $scope.format = 'dd-MM-yyyy';
    $scope.altInputFormats = ['M!/d!/yyyy'];

    $scope.dateOptions = {
      formatYear: "yyyy",
      minDate: startDate,
      maxDate: endDate,
      startingDay: 1
    };

    $scope.formatDateModal = function() {
      console.log($scope.logVariables)
    }

  }]);

<script src="//ajax.googleapis.com/ajax/libs/angularjs/1.6.1/angular.js"></script>
<script src="//ajax.googleapis.com/ajax/libs/angularjs/1.6.1/angular-animate.js"></script>
<script src="//ajax.googleapis.com/ajax/libs/angularjs/1.6.1/angular-sanitize.js"></script>
<script src="//angular-ui.github.io/bootstrap/ui-bootstrap-tpls-2.5.0.js"></script>
<link href="//netdna.bootstrapcdn.com/bootstrap/3.3.7/css/bootstrap.min.css" rel="stylesheet">
<div ng-app="dateExample">
  <div ng-controller="ExampleController">
    <div class="row">
      <div class="col-sm-3 col-md-3">
        <label for="sel1">Label 1</label>
        <p class="input-group">
          <input type="text" class="form-control" uib-datepicker-popup="{{format}}" ng-model="logVariables.startDate" ng-change="formatDateModal()" ng-model-options="{timezone: 'UTC'}" is-open="popup1.opened" datepicker-options="dateOptions" close-text="Close"
            alt-input-formats="altInputFormats" />
          <span class="input-group-btn">
<button type="button" class="btn btn-default" ng-click="open1()">
<i class="glyphicon glyphicon-calendar"></i></button>
</span>
        </p>
      </div>
      <div class="col-sm-3 col-md-3">
        <label for="sel1">Label 2</label>
        <p class="input-group">
          <input type="text" class="form-control" uib-datepicker-popup="{{format}}" ng-model="logVariables.endDate" ng-change="formatDateModal()" ng-model-options="{timezone: 'UTC'}" is-open="popup2.opened" datepicker-options="dateOptions" close-text="Close" alt-input-formats="altInputFormats"
          />
          <span class="input-group-btn">
<button type="button" class="btn btn-default" ng-click="open2()">
 <i class="glyphicon glyphicon-calendar"></i>
</button>
 </span>
        </p>
      </div>
    </div>
  </div>
</div>